Grwusl, usny rcpmoement of the CUIIVBIIWIII~.I t i a ~ s ~ y m in TCC cracking installations with modified silica-alumina cracking catalyst (Durabead 1) leads to a gradual decrease in catalyst attrition. Figure 7 shows observations on one TCC unit (Eastwood and Phalen, 1959). Similar decreases in catalyst attrition and residual coke core frequency resulted .. . . in all other TCC: installations. ~
~~
Generd Appliccability of Altrition Phenomenon
The same sequence of phenomena and resulting deterioration of solid catalyst particles has been demonstrated in other cyclic catalyst operations. A striking example was encountered during use of 3/8-inch diameter chromia-alumina pellets in a cyclic operation involving naphtha aromatization and air regeneration of several hundred cycles. Many pellets fractured. Whole pellets removed and split revealed concentric areas of various colors, and in several instances entire “nuclei” corresponding to cores literally fell out when the shell was split (Figure 8). It is apparent that the demonstrated mechanism of particle ,hell-like alumina
D m e increaseu ainuslvny is seen co result in an upward shift of about 30‘C. of any equivalent temperature point T, (characteristic for any arbitrary condition of radial uniformity in the burning kinetics). The result may also be expressed in terms of a twofold increase in carbon burnoff capacity before reaching equivalent burnoff conditions in diffusion-limited kinetics (see formula 6 of Weisz and Goodwin, 1962).
de.tariorn,tion. atrnnelv linked t,o mnra-tra.nannrt, effects in the
fiastwooa, u. L., rnaren, I. E,., wan Annual meeaing, western Petroleum Refiners Association, San Antonio, Tex., March 16-18, 1959.
Schwartz, A. B., U. S. Patent 2,900,349 (Aup. 18, 1959). Weisz, P. B., 2. physik. Chem. (Frankfurt) 11, 1 (1957). Weisz, P. B., Schwarts, A. B., J . Catalvsis 1, 399 (1962). Weisz, p, B., Goodwin, ?., , J . catalysis2, 397 (1962), Weisz, P. B., Goodwin, R. D.,J. Catalysis 6,227 (1066). RECErmn for review January 15, 1969 ACCEPTEDMarch 24, 1969
FREE CONVECTION AT LOW PRANDTL NUMBERS I N LAMINAR BOUNDARY LAYERS D U D L E Y A.
SAVILLE
Departmen2 of Chemzcal Engineering, Princeton University, Princeton, N . J .
S T U A R T W.
CHURCHILL
School of Chemical Engineering, University of Pennsylvania, Philadelphia, Pa. 19104 A two-term expansion is derived for the Nusselt number for laminar, natural convection about on isothermal, horizontal cylinder immersed in a fluid of vanishingly small Prandtl number. The analysis is carried out for round-nosed cylinders but may be applied to cylinders of other shapes simply by reinterpretation of the independent variables. A numerical result is derived far a circular cylinder and used together with prior theoretical results for Pr 0.7 and m to construct an interpolation farmula for all Pr. This formula agrees well with experimental and theoretical results for Pr 0.01. It i s concluded that one- and two-term expansions of the type previously developed by the authors are adequate for small as well as large Prandtl numbers.
DIMENSIONALanalysis by the method of Hellums and
Churchill (1964) of the partial differential equations describing natural convection ahout a n isotherm1 object immersed in a low Prandtl number fluid under the idealizations of boundary-layer theory yields Nu
a
Gr“‘ Pr’lZ
T o determine the proportionality constant, solutions to the
partial differential equations and boundary conditions are frequently developed in terms of infinite series. Determination of the coe5cients in these series requires the solution of associated ordinary differential equations. The Blasius series as developed by Chiang and Kaye (1962) and the “wedge method’’-the series developed by Meksyn (1961 ) and employed by Acrivos (1962)-are examples for objects which do not admit similarity transformations. Saville and VOL.
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329
Churchill (1967) recently proposed yet another series. All of these procedures have features in common, one being that the first terms in the expansions are fixed by the solution of the same ordinary differential equation. The difference a t this level lies in the interpretation of the independent variables. With the expansions described by Saville and Churchill one term suffices to represent the essential character of free convection around circular cylinders a t Prandtl numbers equal to or greater than 0.7. Indeed, calculations carried out with a two-term expansion for a Prandtl number of 0.01 suggested that one- or two-term representations might be adequate even for small Prandtl numbers. Two-term approximations are therefore developed herein. An important part of the development is the use of asymptotic methods to evaluate certain integrals that arise in the derivation. Outline of Mathematical Model
The development of the mathematical model used herein for free convection is described by Saville and Churchill (1967). I n summary, temperature and stream functions are derived and the temperature function is used to calculate the local Nusselt number according to
NU = - Gr1/4€)y (z, 0)
F ( 4 , ~ ) . The rational numbers KO and p depend only on the class of the body shape-e.g., KO= 4 and p = 1 for sharp-nosed cylinders, and KO= $ and p = $ for round-nosed cylinders. Thus the effect of the body contour is explicitly associated with a nonlinear term in Equation 8, the equation for the modified stream function. This is to be contrasted with the form obtained using the Blasius series. There the effectof the body contour appears associated with the temperature term in the analog of Equation 8 (Chiang and Kaye, 1962). If the Prandtl number is set to zero, the temperature can be taken t o be unity in Equation 8, which then describes a flow near the object analogous to forced convection, driven by an as-yet-unspecified outer flow. This behavior near the object must be matched with the behavior of the stream and temperature functions far from the object. In order to accomplish this Equations 7 and 8 are transformed according t o the scheme T-+ T , F-+ Pr-’/*E, q-+ Pr-l/*p (12) and the viscous term is dropped by letting the Prandtl number approach zero as suggested by Lefevre (1957). Then the outer flow is described by solutions of BEIEPTf - E f T , } - ET, = TPP
(2 )
where € (z, I 0) depends on the Prandtl number. Instead of the independent and dependent variables used by Ostrach (1964) and others to describe free convection, the following independent variables,
BE{EpEfp - E t E p p } - E E p p
(3 )
0
and 71 = ($)1/4Y4-u4[,cj( z ) ~ 1 / 3
(4 )
where S(z) is the sine of the angle between the body force vector and a normal to the surface of the object, and dependent variables $(z,
Y) = (9)3’4E3’4F(E, 71)
(5)
and
e(z, Y) =
(6 )
T(E,71)
are introduced. The appropriate partial differential equations then become
&’{F,Tg - FtT,] - F T , = Pr-lT,,
(7 )
and
94:(F,Ft, - FcF,,} - FF,,
+W(E)FnFn
= T
+ F,,,
T
(14)
T(C-,O)= 1 (15)
= T ( 5 , a ) = 0,
An approximate solution is now sought such that the temperature gradient a t the surface can be calculated. In the determination of the temperature gradient it is convenient to use an integral equation equivalent to Equation 13 along with series representations (in terms of p ) for T (6,p ) and E @ ,P I . Approximate Solutions
An integral equation for T (E, p ) equivalent to Equation 13 and its boundary conditions is T ([, p ) = 1 f
p
e4(Eja)& ([) da -I-
0
l
e-c(t,u)H (4, a)dcr (16)
where G ( ~ , P ) =r E ( E , a ) &
(8)
where
QK (OEpEp =
along with E(E,O) = E,([,
E = / 2 [ ~ ( 4 ~ 1 / 3 dcr
(13)
and
(17 1
0
and
H (E,p ) =
/’
e ~ ( t s a ) +E~~( T ,
E.T~} dcr
(18)
0
The corresponding boundary conditions are T ( 4 , 0 ) = 1, F ( t , O ) = F,(E,O)= F , ( L m ) = T ( E , m ) = O (10)
K ( 0 , termed the principal function by Gortler (1957), determines the explicit influence of the body contour. Since the principal function can be expanded as m
K
(E)
=
C KjEjp
Po
(11)
expansions of similar form are employed for T ( E , q ) and 330
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FUNDAMENTALS
and /31 ( E ) is the gradient of T (E, p ) a t the surface. To solve Equation 16 an expression for E ( E , p ) is required and usually this means that Equation 14 must be solved simultaneously with Equation 16. However, attention may be directed primarily to &(E) rather than to a complete specification of T (E, p ) and E (4, p ) . Assuming that E (4, p ) is known, the determination of PI([) requires the solution of an integral equation. Such a determination would be facilitated if the exponential involved a “large parameter.” However, as Meksyn (1961) has already demonstrated in connection with his studies of forced convection, the behavior of the function G (E, p ) is also important in determining the various integrals.
For a circular cylinder Equation 9 gives R1 = -0.11547, so
Series such as
that
(E) and
-0.6762
+ 0.03716$3/2
(26)
Therefore it follows from the definition of the dimensionless variables that
may be used to determine the value of PI ( E ) if the expansions of the relevant integrals converge rapidly enough. In the present instance 011 ( E ) and a2 ( E ) can be specified [in terms of PI(()] by inspection, rather than by solving (simultaneously) another integral equation. As a consequence G(E, p ) is specified up to terms of order O ( p 3 ) a priori. It appears, now, that in some instances the values of the integrals involved in Equation 16 can be approximated accurately by considering just such truncated expressions as Equations 19 and 20. For example, in the case of a flat plate the exact value of &(E) given by Lefevre (1957) is -0.645, while the use of a two-term expansion for G (5, p ) and Laplace’s method (Erdelyi, 1956) to estimate the value of the integral yields -0.67. The procedure is now applied to a curved body. The relationships between al([) and az(E) on the one hand and P I ( [ ) on the other are found from Equation 14 to be
dcv2
QEai- - aiaz
+ #K (E)aiaz= &/2
~[~CY~]-’&YZ
Nu
(281
= 0.504 Gr1I4Pr1/2
with the radius as the characteristic length. (The first term of the expansion gives a coefficient of $0.548 and the second term contributes -0.042, an 8% correction.) Equations 25 and 27 apply to other round-nosed cylinders. The radius of curvature a t the lower stagnation point is a convenient choice as the characteristic length in the construction of an analog to Equation 28. Generalization of Boundary Layer Solution for Natural Convection
Nu Gr-1/4 Pr-1/4 = f ( P r )
(22 )
+
The fourth and fifth terms on the left of Equation 23 arise solely from the fact that the body contour does not admit of a similarity transformation. In the construction of Equation 23 all of the terms of G ( ( , p ) and H ( E , p ) up to order O ( p 3 ) have been included. Although Equations 21, 22, and 23 could be solved numerically, a series method of solution can also be used. Thus
...
R(O=Ko+KiE’’+
-
-
dE Finally, by employing Laplace’s method along with the series to evaluate the required integrals, a relation for P I ( $ ) is obtained :
+ @1di2[2~1]-112-
The mean Nusselt number averaged over the circumference of the circle is, for the two-term expansion,
A convenient form for expression of the results for the circular cylinder derived here for low Pr together with prior results for larger Pr is
an equation of the Bernoulli type, and
1
-
(29 )
where
[
f ( P r ) = 15.5
+
+ 27.8 Pr ]m
Pr 28.6 Pr1lZ
(30)
The general form of f ( P r ) was suggested by the work of Lefevre (1957) for sharp-nosed bodies. The results herein for Pr- 0 (Equation 28), the numerical solutions of Saville and Churchill (1967) for Pr = 0.7, and the numerical results of Lefevre (1957) for P r + m were used to fix the three coefficients in Equation 30. Equations 29 and 30 yield results in good agreement with the experimental data of Jodlbauer as reported by Schlichting (1968) for a Prandtl number of 0.7, and by Schutz (1963) for a Prandtl number of 1760, as well as with the correlation of McAdams (1954). For a Prandtl number of 0.01 Equation 30 gives a value for f ( P r ) of 0.0482. This value can be compared with the numerical solution of the boundary layer equations by Saville and Churchill (1967) for the first term in which is 0.048, and a correlation of experimental data by Hyman, Bonilla, and Ehrlich (1953) which gives a value of 0.045. Conclusions
+
+
pi ( E ) = ~1‘0) P~(’)E’
. J
Because of the form of Equations 21, 22, and 23, K1 can be scaled out of the equations so that a general solution applicable to any round-nosed cylinder can be obt.ained. Thus
One or two terms of the series representations introduced by Saville and Churchill (1967) are adequate to represent the Nusselt number for horizontal circular cylinders for small as well as large Prandtl numbers; and the calculation of approximations for small Prandtl numbers according to the method developed herein is both straightforward and fairly accurate, a t least as far as can be judged from the available theoretical and experimental values. Nomenclature
E (E, p ) f(Pr)
F (E, q )
= Pr1/2F ( E , q ) , scaled stream function, Equation 12 = interpolating function defined by Equation 30 = (5,y ) (Q5)-3’4, a modified stream function Equa-
+
tion 5 VOL.
8
NO.
2
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1 9 6 9
331
= acceleration due to gravity, (length)/
9
G(E, p )
=
/I
7
0 (2,y )
= g@PAB/v2,Grashof number
/
H ( 6 ,P )
=
e e, 68
r n
CG(4, CY)I%[E~T, - E O , T ~dol, l Equation
0
18 k = thermal conductivity, (energy)/ (time) (length) (temperature) = principal function, Equation 9 K (6) KO,K1 = coefficients in expansion of principal function, Equation 11 1 = characteristic length = ql/kAO, Susselt number Nu E = Ql/kAO, average Kusselt number Pr = V / K , Prandtl number q = local heat flux density at surface, (energy)/ (length)2 (time) Q = average heat flux density, (energy)/ (length)2 (time) S ( X ) = sine of angle between body force vector and a normal to surface of immersed object !!‘(E, 7 ) = O(z, y ) , Equation 6 X = xl/l, dimensionless distance along surface measured from lower stagnation point 21 = distance along surface, length Y = y1 Gr114/l, boundary layer coordinate normal to surface Y1 = distance normal to surface, length a1( E ) = coefficient in the expansion of E (Ej p ) , Equation 19 al(O), = coefficients in the expansion of a1 ( E ) , Equation 24 a2 ( E ) = coefficient in the expansion of E (E, p ) , Equation 19 az(’), al(l)= coefficients in the expansion of a2 ( i ) ,Equation 24
P
= coefficient of thermal expansion, (temperature)-’
/31(i)
=
coefficient in the expansion of T ( i , p ) ,Equation 20
= coefficients in the expansion of /31 (E), Equation 24 = ($)‘14y[S (z)]li3[ (z)-lI4, scaled normal coordi-
nate, Equation 4
E ( [ ,a) da, Equation 17
0
Gr
&@),
A0
(e
= - O,)/AO, dimensionless temperature = local temperature = surface temperature = temperature far from surface = characteristic temperature difference
(e, em),
V
thermal diffusivity, (length)2/time = 7 Pr1IZl scaled normal coordinate, Equation 12 = exponent in the expansion of the principal function, Equation 12 = kinematic viscosity, (length)2/time
4
=
=
K
P P
II. (x,y )
/d
[ S ( ( U ) ] da, ~ ’ ~scaled tangential coordinate, Equation 3 = Lagrange’s two-dimensional stream function
Literature Cited
Acrivos, A,, Chem. Eng. Sci. 17, 457 (1962). Chiang, T., Kaye, J., Proceedings of 4th National Congress Applied Mechanics, Berkeley, p. 1213, 1962. Erdelyi, A., “Asymptotic Expansions,” p. 36, Dover, New York, 14.56
Girire;, H., J . Math. Mech. 6, 1 (1957). Hellums, J. D., Churchill, S. W., A.Z.Ch.E. J . 10, 110 (1964). Hvman. S. C.. Bonilla. C. F.. Ehrlich. S. W.. Chem. Ena. Proar. Lef&& E. J , Proceedings‘ i f 9th International Congress of Applied Mechanics, Brussels, Vol. 4, p. 168, 1957. hlchdams, W . H., “Heat Transmission,” 3rd ed., PvfcGraw-Hill, New York, 1954. Meksyn, D., “New Methods in Boundary Layer Theory,” Pergarnon Press, ?Jew York, 1961. Ostrach, S., “Laminar Flows with Body Forces” in “Theory of Laminar Flows.” F. K. Moore, ed., Princeton University Press, Princeton, N. J., 1964. Saville, D. A,, Churchill, S. W., J . F l m d Mech. 29, 391 (1967). Schlichtmg, €I., “Boundary Layer Theory,” 6th ed., p. 305, McGraw-Hill, New York, 1968. Schutz, G., Intern. J . Heat Mass Transfer 6, 873 (1963). RECEIVED for review January 14, 1969 ACCEPTEDFebruary 7, 1969
SUFFICIENT CONDITIONS FOR STABILITY OF FLUID MOTIONS CONSTITUTIVELY DESCRIBED BY THE I N F I N I T E S I M A L THEORY OF VISCOELASTICITY MARTIN R. FEINBERG‘ AND WILLIAM R. SCHOWALTER Depnrtmeiat of Chemical Engineering, Princeton I’niversity, Princeton, IT. J . 08640 An energy method is employed to determine sufficient conditions for the stability of fluid motions which are constitutively described by the infinitesimal theory of viscoelasticity. The analysis is restricted to situations for which the fluid is either totally confined by solid walls or the velocity field for the disturbed fluid motion may b e assumed to b e periodic along an axis of symmetry. The sufficient conditions far stability established appear to be somewhat conservative. For the conditions to yield nontrivial information, the fluid memory must have a Newtonian aspect to it such that emphasis must be placed on events occurring a t the present time.
THE literature describing problems in hydrodynamic sta-
bility consists, for the most part, of solutions of thelinearized momentum equation for Newtonian fluids, I n this paper we presentaddress, Department of Chemical Engineering, Uni-
versity of Rochester, Rochester, N. Y. 14627. 332
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FUNDAMENTALS
avoid both of these restrictions. The analysis presented here leads to conditions sufficient to guarantee stability of several fluid motions FThich are described constitutively by the infinitesimal theory of linear viscoelasticity. A radical departure from the linear theory was presented by
