C1, C2, C3 = integration constants, eq 24 Db = effective bubble diameter, cm D, = average particle size, cm D, = reactor diameter, cm D, = molecular diffusion coefficient, cm2/s g = gravitational acceleration, 980 cm/s2 k , = mass transfer coefficient, g-mol/s cm2 atm Kbc = interchange coefficient between bubble phase and cloud phase, based on volume of bubble, l/s K,, = interchange coefficient between cloud-wake and emulsion, based on volume of bubble, I/s L = total height of fluidized bed, cm 41, q2, q3 = eigenvectors of matrix A , eq 24 R = gasconstant 'ub = bubble velocity, cm/s u, = cloud-wake velocity, cm/s ue = gas velocity through emulsion phase, cm/s u,f = minimum fluidization velocity, cm/s uo = superficial gas velocity, cm/s u s = downward velocity of solids in emulsion phase, cm/s y = average concentration of napththalene in gas stream, mole fraction yb = concentration of naphthalene in bubble phase, mole fraction y c = concentration of naphthalene in cloud-wake phase, mole fraction y e = concentration of naphthalene in emulsion phase, mole fraction y s = concentration of naphthalene a t surface of active particles, mole fraction z = height within fluidized bed, cm
Greek Letters = volume fraction, volume of cloud-wake phase to volume of bubble phase yc = volumetric fraction of solids in cloud-wake phase ye = volumetric fraction of solids in emulsion phase 6 = bubble fraction, ( u g - Umf)/ub tc = void fraction of cloud-wake phase CY
t,
= void fraction of emulsion phase
emf = void fraction of bed at minimum fluidization conditions
= constant, YcUbt,f/(l - tmf) = constant, Yeuetmf/(l- emf) XI, 12, A3 = eigenvalues of matrix A 17 = normalized bed height, z / L 5 = average normalized concentration, ( y , - y ) / y , &, = normalized concentration of naphthalene in bubble phase, (Y - Y b)Y s Ec = normalized concentration of napthalene in cloud-wake phase, (Y, - Y,)/Y, [, = normalized concentration of naphthalene in emulsion phase, ( Y , - Ye)/Ys E = solution vector, eq 18 ~1 ~2
L i t e r a t u r e Cited Chavarie, C., Grace, J. R.. lnd. Eng. Chem., fundam., 14, 75 (1975). Davidson, J. F.. Harrison, D., "Fluidized Particles", Cambridge University Press, 1963. Danckwerts. P. V., Chem. Eng. Sci.. 2, 1 (1953). Higbie, R.. Trans. Am. hst. Chem. Eng., 31, 365 (1935). Hsiung. T. H.. PhD. Dissertation, Northwestern University, Evanston, Ill., 1975. Kato. K., Ito, U.. J. Chem. Eng. Jpn., 7, 40 (1974). Kato, K., Wen, C. Y., Chem. Eng. Sci., 24, 1351 (1969). Kunii, D., Levenspiel, O., lnd. Eng. Chem., fundam., 7, 446 (1968). Kunii, D., Levenspiel, 0.. "Fluidization Engineering", Wiley, New York, N.Y., 1969. May, W. G., Chem. Eng. Prog.. 55,49 (1959). Mori, S..Wen, C. Y., "Estimation of BubbleDiameter in Gaseous Fluidized-Beds". presented at the 67th Annual Meeting, American Institute of Chemical Engineers, Washington, D.C., Dec 1-5, 1974. Pyle, D. L., Adv. Chem. Ser., No. 109, (1972). Tellis, C. B., Hulburt, H. M., Longfield, J. E., "First Pacific Chemical Engineering Congress", Part II, Kyoto, Japan, p 199, 1972. Wilde, D. J., Beightler, C. S., "Foundations of Optimization". Chapter 6.PrenticaHall, Inc., Englewocd Cliffs, N.J., 1967. Wilkins, G. S.,Thodos, G., A.l.Ch.E. J., 15, 47 (1969). Yoon, P., Thodos, G., Chem. Eng. Sci., 27, 1549 (1972). Yoshida, K., Kunii, D., J. Chem. Eng. Jpn.. 1, 11 (1968).
Received for review April 24,1975 Accepted May 6,1976
E XP E R IMENTA 1 TECHNIQUES
Rheology of Power Law Fluids William A. Hyman Bioengineering Program, Texas A&M University, College Station, Texas 77843
A numerical technique is presented for the treatment of data obtained with a cylindrical viscometer to characterize a power-law fluid. The method is an adaptation of standard curve fitting techniques and provides an alternative to graphical methods, in which individual judgment becomes a factor. The method also provides a statistical estimate of confidence intervals in the two power-law parameters.
Introduction The difficulties inherent in measuring non-Newtonian viscosities with industrial rotational viscometers are well recognized (McKennel, 1960). These instruments commonly rotate a cylindrical spindle (or bob) in a concentric container
of fluid (the cup), or in an effectively infinite reservoir of fluid. The spindle is rotated a t constant angular velocity and the concomitant torque on the spindle is measured. The difficulty is due primarily to the inability to predict the shear rate on Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
215
Table I. Values of
similar viscometer is readily obtained as illustrated in the s a m d e calculation to follow. In this method the data are used to generate the following sums.
t,/2
No. of data pointsa
a = 0.01
LY
= 0.05
M
M 3 4
63.657 9.925 5 5.841 6 4.604 7 4.032 8 3.707 9 3.499 10 3.355 Equivalent to M - 2 degrees of freedom.
i=l
12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306
M i=l
s3
=M
M
M
i=l
S=
SlSZ - s 3 2 M ( M - 2)S2
T= the bob of the instrument which corresponds to the measured torque. In the case of Newtonian fluids this shear rate is related only to geometric factors and bob angular velocity. With non-Newtonian fluids, however, the deformation properties of the fluid are also important in determining the shear rate. The approaches to this problem have included new instrument designs such as a multiple spindle Couette viscometer (Eubank and Fort, 1967) and a continuous flow pipeline system (Scheve et al., 1974). New data reduction methods have also been proposed (Kreiger and Maron, 1952; Back, 1959; Rosen, 1971,1972). The latter are most useful when the general form of the shear rate-shear stress diagram is known in advance. Rosen (1971, 1972), for example, has presented graphical methods appropriate for determining the two parameters in power law representations of non-Newtonian behavior. In such a representation the shear stress-shear rate relationship takes the form
M
- ix 1% wi log ~i =l i=l
1log wi log ~i
1
isl
1
M
Q =-
M
log 7 i
log wi
M i = 1
The constants in (1) can then be determined from
n = -s3 s 1
and
k=
(;)n
x
10T-nfl
The accuracy of the fit obtained by this process can be assessed by computing confidence intervals for n and k . The half range in n , for a level of confidence a,is given by (10)
(1)
=kyn
These methods require best fits “by eye” between the experimental data and appropriate templates. While this technique is useful it contains the possibility of human variation affecting the results and also does not provide a check on the reasonableness of the determinations. An alternative procedure is the numerical treatment of the data presented here which yields the two parameters of the power-law representation in a systematic way which is free from individual judgment and, in addition, provides a check on the reliability of the results. As formulated here, it is applicable when the radius of the cup is large compared to that of the bob. Method It is assumed that measured values of the shear stress are available at given angular velocities of a cylindrical bob in an effectively infinite reservoir of fluid. The comparable case of having measured scale values a t given rpm settings such as would occur when using a Brookfield Synchro-Lectric or
This expression is interpreted to mean that the value of n is within the range n f 6, with a level of confidence a.The required level of confidence is chosen by the user on the basis of the accuracy desired. The range which is computed increases as the level of confidence decreases since small values of a are interpreted as a high probability of the actual value of n lying in the indicated range. This can be seen by considering the modified “ t ” table given in Table I. If the range computed is very large then the goodness of fit is poor and the fluid in question may not be of the power-law type. The half range in k for a level of confidence a is given by dk
= t,/2kS
1/” s 2
+ s1+ ( M Q ) 2(1 + log MS1
Therefore k lies in the range k f 6 k with a level of confidence a.This numerical procedure is most easily performed when the tabular form given in Table I1 is used. The quantities in eq 2-7 can then be easily computed.
Table 11. Tabular Computation Form Data
Totals 216
Ind. Eng. Chem., Fundam., Vol.
Data
-
zlog wi 15, No. 3, 1976
z(log W i ) 2
(11)
log T i
z (log T i ) 2
zlog wi log Ti
where L is the effective length of the bob and K is the instrument spring constant. In the present case R = 0.942 cm, L = 7.88 cm, and K = 674 dyn-cm. Substitution of this expression and the numerical values into eq 16 and rearrangement yields
Table 111. Sample Data rPm
Dial reading
6
20.1 28.0 44.
12
30 60
63.5
where
Theoretical Development The spatial variation of the shear stress component in a fluid in a cylindrical flow driven by the rotation of a cylinder is given by
k* = 6.52K (19) Table IV shows the computational format. From these values and eq 2-7
Si = 4(7.115) - (5.112)* = 2.327 S2
R2
7=-7*
r2
S=
For the axisymmetric cylindrical geometry of interest here d
V
z(T)
(14)
Combining (13) and the integral of (14) it can be shown that =k
(21)
2.327(0.581) - (1.158)’ = o.00237 4(2)(0.581) T = -6.196 - 1.549 4 5.112 - 1.278 Q=-4
=k
(16)
so that “a” becomes k ( 2 / ~ zand ) ~ “b” is the index n itself. The standard procedures then yield estimates of “a” and “b”. These give n and, in turn, allow calculation of k. The bounds on “ a ” and “b” in the standard formulation are estimated in terms of the “t” distribution. In order to find the bounds of k the comparable bounds on “ a ” and “b” must be combined which results in (11).This is also within the scope of standard practice (Beers, 1957).
(22) (23)
On the basis of the computed quantities the exponent n is then computed from eq 8
(f)n
The statistical treatment of the data in obtaining a best fit to (15) follows the standard methods (Miller and Freund, 1965) for the least-squares best fit of data by equations of the type y = ax b. In order to employ these methods here (15) can be regrouped to read T
= 4(9.743) - (6.196)2 = .581
S3 = 4(8.208) - (5.112)(6.196) = 1.158
Combining (1)and (12)
=
(20)
n=--1.158 - 0.498 2.327 The confidence interval on n is calculated using eq 10. With (Y = 0.01 and 4 trials t a / 2 = 9.925. Then 6, = 9.925
+ (5.112)2) V’0.00237(2.327 4(2.327)
or 6 , = 0.085
The constant k * is obtained from eq 9 with 1 5 d x replacing n/2. Therefore
Then from eq 19 Sample Calculation The following example is based on the data given in Table I11 which was obtained with a Brookfield Synchro Lectric viscometer with the LV-1 spindle (Rosen, 1971). It is convenient to recast eq 16 into a form in which the raw data of scale reading, T * , and rpm, w * , can be used directly. This can be done by noting that 7 =
(29)
The confidence interval on h * is calculated using eq 11 6k*
= 9.925(0.00237)12.583
T*K 200aR2L
(30)
Table IV. Tabular Computation Form for Sample Calculation Data
Data -
Trial no.
w*
log w *
(log w * ) *
T*
log T*
(log T * ) 2
log w * log T*
1 2 3
6 12
0.778 1.079 1.477 1.778
0.606 1.165 2.182 3.162
20.1 28.0 44.0 63.5
1.303 1.447 1.643 1.803
1.698 2.094 2.701 3.250
1.014 1.561 2.427 3.206
5.112
7.115
6.196
9.743
8.208
4
30 60
Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
217
The corresponding confidence interval on k itself is 0.89 6.52
&=--=
0.14
Conclusions This numerical treatment offers a method for reducing viscometer data obtained for a power-law fluid. It provides estimates and confidence intervals for the two necessary parameters which are nonsubjective, which is especially important in communicating rheological measurements. The general method presented here can be specialized to particular instruments by modifying the computational scheme so that it deals directly with the recorded d a t a torque (or scale factor) and rpm. This can readily be accomplished by appropriate substitution in eq 16. If all additional constants ~ , subsequent procedure are then grouped with k ( 2 / r ~ )the becomes clear. Nomenclature k,n = power law constants K = instrument spring constant L = bob effective length M = number of data points r = radial coordinate R = bobradius S, SI,S2, SB,T = sums as defined in eq 2-6
T* = instrument dial reading t 4 = value of t distribution for level of confidence and M - 2 degrees of freedom V = tangential velocity Greek Letters = level of confidence y = shearrate 6 k , 6, = half ranges in k and n 7,7* = shear stress, shear stress on bob w = bob angular velocity w* = bobrpm ~ i , = pairs of data points of bob shear and angular velocity i2 = sum as defined in (7) CY
Literature Cited Back. A. L., Rubber Age, 84, 639 (1959). Beers, Y., "Introduction to the Theory of Enors", p 26, Addison-Wesley, Reading, Mass., 1957. Eubank, P. T., Fort, B. F., /SA Trans.,6, 298 (1967). Kreiger. J. M., Maron, S. H., J. Appl. Phys., 23, 147 (1952). McKennel, R.. Anal. Che?, 32, 1458 (1960). Miller, I., Freund, J. E., Probability and Statistics for Engineers", p 226, Prentice-Hall, Englewood Cliffs, N.J., 1965. Rosen, M. R., J. Colloid lnterface Sci., 36, 350 (1971). Rosen. M. R., J. Colloidlnferface Sci., 39,413 (1972). Scheve, J. L., Abraham, W. A,, Lancaster. B. B.. lnd. Eng. Chem., Fundam., 13, 150 (1974).
Received for review November 21,1974 Accepted February 16,1976
A Stirred Flow Microbalance Reactor for Catalyst Studies F. E. Massoth* and S. W. Cowley Depaltment of Mining, Metallurgical and Fuels Engineering, University of Utah, Salt Lake City, Utah 84 I12
A stirred flow microbalance reactor was constructed for simultaneously measuring catalyst weight changes and activities. It consists of a heated glass reactor in which the catalyst, suspended from a microbalance, is surrounded by a squirrel cage stirrer. Gas flow is up the stirrer shaft, into the bottom of the reactor volume, and out the reactor top. Catalyst weight changes are monitored continuously with a Cahn recording balance and catalyst conversions determined by gas chromatographic analysis of the reactor effluent stream. Gas mixing and gas-tocatalyst mass transfer efficiency tests demonstrated that the reactor performed as a constant stirred tank reactor. Consequently, direct catalytic reaction rate data are obtained. In addition, weight changes attending reaction under various steady-state conditions directly reflect amounts of equilibrium adsorption species present. An example of its use in a kinetic study of the hydrogenation of butene-1 over a molybdena-alumina catalyst is presented.
Introduction One of the more useful tools in kinetic investigation of catalytic reactions is the constant stirred tank reactor, as exemplified by the Carberry reactor (Carberry, 1964). The advantages of this type of reactor over the more conventional fixed-bed type are that rate data are directly obtained and better isothermal temperature control is achieved. Useful information on characterization of catalysts can be obtained by use of flow gravimetric techniques (Massoth, 1972).Among other things, this technique permits: (1)in situ pretreatment to a known level, (2) monitoring of amounts of a poison added to a catalyst, (3) following catalyst coking during reaction, and (4) determining changes in adsorbed species with reaction conditions. In order to combine the advantages of both techniques in the same reactor, a constant stirred, flow micro218
Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
balance reactor (SFMBR) was developed. With it, catalytic reaction rates and catalyst weight changes can be simultaneously followed. The only apparatus comparable to that described herein is that reported by Hsu and Kabel(1974), who used simultaneous gravimetric analysis with a static batch reactor. Their apparatus gives somewhat different information than ours in that reactant and product concentrations change with time whereas in our reactor steady-state concentrations are invariant in real time.
Experimental Section The flow microbalance has been previously described (Massoth, 1972). To simulate a constant stirred tank reactor, the assembly was modified to incorporate a stirrer. Details are
