Estimation of Model Parameters Using Quasilinearization SIR:Ramaker, Smith, and lIurrill(l970) discuss the application of quasilinearization to the determination of numerical values of parameters in a dynamic model. Their efforts in studying the effect of various factors on convergence and attempts a t increasing the convergence region represent a valuable contribution in the area of parameter estimation. However, a few points in the above article require clarification. Application of the quasilinearization algorithm to the following system of nonlinear equations d? - = J(?,t) dt
which is their Equation 8. They then state that the coefficients k initial conditions for the differential equations if they were available. This is true only for the special case in which the initial condition for the particular solution vector is the null vector and the initial condition for the matrix of homogeneous solutions is the identity matrix. The authors state that the computational effort can be reduced by defining the state variables in terms of the change in state variables from one iteration to the next p, would be the
6 =
results in the following system of linear equations (Lee, 1968)
where in their matrix
1 1
=
(7)
JJ
which is the first half of their Equation 7 , with the implication that Equation 7 is equivalent to Equation 3. Equation 3 can be written as
then Equation 8 will reduce to Equation 7 and indeed Equation 7 is equivalent to Equation 3. However, from Equation 8
(ad)
6 corresponds to
i, = K,-,JI
I n applying the quasilinearization algorithm to their Equation 4,the authors have made an error in specifying 6, which is the equation above their Equation 7 . The correct specification of 6 is
L
(6)
A’ corresponds to aj
and their vector
- zi.
which is the second half of their Equation 7 . They then present the differential equation
8 where subscript i represents iteration number. The authors express the resulting linear equation-their Equation 6in the form
?t+1
0 0 0
+
(10)
fi-1
Thus f f = Z i only when the iterative process is near convergence-that is, when 8, = 0 and St-, = J t . Thus the authors’ Equation 7 , supposedly quasilinearization in terms of state variable changes, is equivalent t o the standard quasilinearization algorithm, the authors’ Equation 6, only when the iterative process is near convergence. literature Cited
(4)
The authors go on to say that the solution to Equation 3 is (5)
Lee, E. S., “Quasilinearization and Invariant Imbedding,” Academic Press, New York, 1968. Ramaker, B. L., Smith, C. L., RIurrill, P. W., IND.EXG.CHEW FUNDAM. 9, 28 (1970). Gale G. Hoyer The Dow Chemical Co. Midland, Jiich. 48640
Vaporization of Droplets in High-Temperature Gas Streams SIR:Frazier and Hellier (1969) have measured the vaporization of nonsupported Freon-113 droplets into heated air. They state t h a t the correlation of Ranz and Xarshall (1952) underestimates the rnass transfer coefficient by a factor of 4. To test this conclusion, we have recalculated the evaporation rate from the Ranz-Marshall correlation, corrected for transpiration effects. We find a much smaller deviation, 3370, between our predicted transfer rate and that which was observed. This is rather good agreement, in view of t h e uncertainties in the measurement and in our estimates of the physical properties for this system. Our calculation procedure is a n extension of t h a t of Ranz
and Marshall, as indicated below. We use the interfacial energy balance S..IoAR,4o
+ yo(C) +
qo(7)
=
0
(1)
which holds for vaporization of pure X (here, Freon-113) from an isothermal droplet. The fluxes and int’erfacial states are regarded as averages over t’he surface of the droplet. The temperat’ure of the gas a t the int,erface is T oand the composit’ion there is approxiniated as X A O = P A , v,,/p. To start the calculation, a trial value of Tois chosen and t,he resulting interfacial composition, z A 0 , is calculated from vapor pressure data reported by Stull (1947). Then the other Ind. Eng. Chem. Fundom., Vol. 9, No. 3, 1970
515
fluid properties are calculated a t the arithmetic mean fluid state (Tj,xA and inserted into the Ranz-Marshall correlation to obtain the transfer coefficients. We use their correlation as stated by Bird et al. (1960, p. 647), and use the results of the film theory (Equations 21.5-35 and 36 of the same source) to obtain the mean fluxes into the gas phase:
(3)
These expressions include the corrections for the net flow through the interface based on the film theory. The correction to q0(') was omitted by Ranz and Marshall (1952), as it was negligible for their experiments. However, it is important for the present system because of the large temperature difference (more than 600°C). The radiant flux t o a droplet in Frazier and Hellier's experiment comes almost entirely from the nozzle. For purposes of estimating the maximum expected radiant flux, the nozzle opening is taken as a black disk a t temperature T , = 669°C and the droplet as a black sphere. Then one obtains yo(') = uF12(To4 - Tm4)
(4)
where u is the Stefan-Boltzmann constant. The view factor, F12,from the droplet to the disk is 0.22 for a 1.5-inch diameter disk located 0.5 inch away from the sphere. Calculation of N A o ~ R A oqo(c) qo(') for two or more trial values of T o permits an interpolation to find the T O consistent with Equation 1. Linear interpolation gives satisfactory convergence, Our results for Frazier and Hellier's experiment are as follows:
+
+
To = 27.5"C XAO
=
Dvmpj
-
0.473
T,
=
669°C
XAm
=
0.000
(0.044 cm) (375.3
E)
TI XAf
= 348.2"C =
(0.001302
0.236
t) =
78.1
0.000275 cm sec
(0,2215
5)
(0.000275
0.000275
(A) =1 (0,001302
=
) @)
">
cma (0.294 sec
h, = 0.0138 k,,
-
cm sec
cal om2 sec "C
g-moles 0.000884 cm2 sec
C,A =
cal 38.6 g-mole "C g-moles cm2 sec
N A =~ 0.000565 -
516
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
=
0.698
-
-0.24
cal cm2 sec
QO(')
The approach speed, v, = 375.3 cm per second, was obtained vectorially from the droplet velocity (186 cm per second, horizontal) and the air velocity (326 cm per second, vertical) as measured in the laboratory coordinates. The following sources of physical properties were used. The viscosity and binary diffusivity were computed from Equations 1.4-18 to 20 and 16.4-13 of Bird et al. (1960), with the following Lennard-Jones parameters: i,
i
eij/kB,
OK
Qij,A
AI A
339.6
5.90
Bl B
97.0 181.5
3.617 4.62
AI B
Source
Critical properties from D u Pont (1962) and correlation iii of Tee et al. (1966) Hirschfelder et al. (1964) Geometric combination for both C A B and U A B
The thermal conductivity was computed from Equation 8.317 of Bird et al. (1960) by using tabulated values for air from the National Bureau of Standards (1955) and a modified Eucken formula of Hirschfelder (1957) for Freon-113. The heat capacity of air was taken from the National Bureau of Standards (1955); values for Freon-113 were computed by the method of Dobratz (1941) and multiplied by a factor 1.042 to fit the value measured a t 6OoC by D u Pont (1962). Ideal gas behavior was assumed in calculating the density and heat capacity of the mixture. The liquid density and heat of vaporization of Freon-113 as measured by the D u Pont Co. (1962) were corrected to temperature Toby the methods given by Hougen et al. (1954, 1959). It was expected that the evaporation rate would be sensitive to the value used for DAB, the property which we consider most uncertain, However, a recalculation with D A B 5% larger gave a lower droplet temperature but an identical evaporation rate: N A=~0.000566 g-moles/cm2 sec. The evaporation rate was measured by Frazier and Hellier in the form K = -dt/d(D2) = 56 sec/cm2. From this value and a liquid density of 1.56 grams per cc at 27.5"C1we get
= 0.664
(0.0000919 cm sec "C
cal cm2 sec
qo(') = -3.64
NAO(= ~ ~0.00085 ~ ~ ) g-moles/cm* sec This measurement is rather uncertain, as can be seen from Frazier and Hellier's Figures 2 and 3. Furthermore] the duration of exposure of the drops to the hot air (about 12 milliseconds) appears to be far too short t o establish isothermal conditions inside the drops. From these considerations, it appears that the measured and predicted evaporation rates agree within their uncertainty. Further measurements, with longer contact times, would be desirable. Acknowledgment
The computations were programmed by Jan P. Spensen. Nomenclature
6,
= specific heat capacity] cal/(g "C)
DAB
= = =
e, D
molar heat capacity, cal/(g-mole "C) droplet diameter] cm binary diffusivity, cm2/sec
0
= direct view factor, dimensionless
differential heat of vaporization of A at interfacial conditions, cal/g-mole = mean heat transfer coefficient, cal/(cm2 sec “C) = evaporation constant, sec/cm2 = thermal conductivity, cal/(cm sec “C) = mean mass transfer coefficient, g-moles/(cm2 sec) = mean flux of A into the gas phase, g-moles/(cm2 sec) atm P= vapor pressure of pure A at temperature To, = static pressure in approaching fluid, a t m = conductive heat flux into gas phase, cal/(cm2 sec) = net radiant h e t t flux from drop, cal/(cm2 sec) = temperature, C = time from drop formation, sec = approach speed of gas stream relative t o drop, cm/sec = mole fraction of A in gas = Lennard-Jones energy parameter, OK = viscosity, g/(cm sec) = density, g/cc = Stefan-Boltzmann constant, 1.355 X cal/ (cm2 sec OK4) = Lennard-Jones collision diameter, O K =
2k PA, V
t 0,
Uij
~
interfacial conditions approaching fluid conditions = “film” conditions: ‘/z(To TJ,‘ / Z ( X A O 4-X = =
cn
f
+
literature Cited
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, 1960. Dobrata, C. J., Ind. Eng. Chem. 33, 759 (1941). D u Pont de Nemours & Go., Inc., E. l., Wilmington, Del., “Properties and Applications of the Freon Fluorocarbons,” Tech. Bull. B-2 (1962). Frazier, G. C., Hellier, W. W., IND.ENG.CHEM.FCXDAM. 8, 807 (1969).
Hirschfelder, J. O., J. Chem. Phys. 26,282 (1967). Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical Process Princi les,” 2nd ed., Part I, p. 281, 1954, Part 11, p. 586, Wiley, &‘e, York, 1959. National Bureau of Standards, “Tables of Thermal Properties of Gases,” Circ. C564 (1955). Ranz, W. E., Marshall, W. R., Chem. Eng. Progr. 48, 173 (1952). Stull, D. R., Ind. Eng. Chem. 39, 517 (1947). Tee, L. S., Gotoh, S., Stewart, W. E., IND.ENG.CHEM.FUNDAM. 5, 356 (1966).
E. J . Crosby W . E. Stewart
SUBSCRIPTS A B
= =
A ~ ) p,
University Of Wisconsin Madison, Wis. 53706
evaporating species (Freon-113) nontransferred gaseous species (air)
Dynamic Similarity in Continuous Stirred Tank Reactors SIR:A recent communication by Cerro and Parera (1970) discusses residence time distributions in stirred tank reactors. They use a statistic, denoted as ur2, to correlate their experimental results and apparently believe t h a t this quantity provides a superior measure for the degree of mixing. The present note is intended to clarify the physical meaning of u12 and raises the question of whether this quantity is indeed a superior tool for characterizing macromixing. Following Zwietering (1959) and Levenspiel (1962) , function I is the frequency function of the internal age distribution and is given by I(t)
[l
= 1/7.
- F(t)]
(1)
less, but its mean value, 61,has units of minutes. Also, the cumulative distribution function, F , is said to have units of minutes, when it must be dimensionless, regardless of the units for 8. Similarly, I must either be dimensionless or have units of reciprocal t h e . For the present purposes we assume that e is indeed dimensionless and that Equation 5 holds exactly. Then
6, =
where 8 = t , priori from
7.The
mean residence time, 7
=
(2) 7,
is found either a
v/g
J0
[l
- F ( t ) ]dt
u12 =
Lrn(e
- e,)zi(e)de
g12
If scaling has been done using Equation 4,then n-
(5) exactly. If done by 3, the integral in Equation 5 should be close t o unity; and any departure from unity provides a measure of experimental accuracy (assuming no truly dead zones have been included in the definition of V ) . The paper by Cerro and Parera is unclear regarding scaling. The mean residence time, 7, is apparently denoted as 6 with units of minutes. The quantity e is said to be dimension-
(7)
e, = a = 1/2(1 + u’)
and (4)
(6)
is the variance of ages, often denoted as var CY. As first pointed out by Zwietering (1959), these quantities can be related to moments of the residence time distribution. I n dimensionless form
(3)
or directly from the step change response, =
eI(e)de
is seen to be the mean age within the system, often denoted as a. Similarly,
where t has dimensions of time. In terms of a dimensionless time variable,
~ ( e=) 1 - F(e)
Lm
=
var
o(
p3!
= -
3
- (1 ~
+
(8) 4
2
4
(9)
where az is t h e dimensionless variance of the residence time distribution and p3‘ is the dimensionless third moment about the origin. 61and u12are the mean and variance of the internal age distribution, not of the residence time distribution. Table I compares 61and U I ~to various moments of the residence time distribution for the ideal cases of piston flow and perfect mixing. 61,u2, and p ~ are ’ all measures of the second moment of the residence time distribution; and there seems little justification for regarding any one as superior to the are measures of the third others. Similarly, p3, p3’, and moment. Since u12has the range 0.0833 to 1.0, it hardly seems more sensitive to macromixing than, say, p3, which has the Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
5 17
