= input concentration of A = sample average for concentration = initial concentration of A = heat capacity of fluid = energy of activation = heat of reaction = reaction coefficient = preexponential factor = volumetric flow rate of coolant = input volumetric flow rate = qi/V = output volumetric flow rate = heat transfer rate
T) S
t t*
T m
1 i
of A
ideal gas constant reaction rate function = sample standard deviation = time = =
= t/r =
= = = = =
=
temperature of fluid in the vessel inlet fluid temperature initial temperature of fluid overall heat transfer coefficient velocity of flow volume of vessel general random variable (noise) designated by its subscript
GREEKLETTERS Y = coefficient of variation x = dimensionless flow rate J ! = dimensionless inlet concentration 11 P
= =
T ( P C ~ ) A /AH(CAJ) (-
fluid density
E v r
= CA/(CAi) = dimensionless inlet = (qi)t/V
temperature
SUBSCRIPTS C
f
= =
1
=
0
=
coolant output inlet initial
OVERLAY
A
OTHER ( )
= =
sample mean estimated
=
expected value
literature Cited
Ark, R., Amundson, N. R., Chem. Eng. Sci. 9, 250 (1958). Berryman, J. E., Himmelblau, D. R‘I., Ind. Eng. Chem., Process Des. Develop. 10,441 (1971). Homan, C. J., Tierney, J. W., Chem. Eng. Sci. 12, 153 (1960). King, R. P., paper B 4-35 presented at the Second International Symposium on Reactor Engineering, Amsterdam, May 2-4, 1972, Elsevier, Amsterdam, 1972. Krambeck, F. J., Katz, S. Shinnar, R., Combust. Sci. Technol. 4, 221 (1972). Laning, J. H., Battin, R. H., “Random Processes in Automatic Control,” McGraw-Hill, New York, N. Y., 1956. Mortensen, R. E., J . Statis. Phys. 1, 271 (1969). Pell, J. M., Aris, R., IND.ESG.CHEM.,FUNDAM. 8, 339 (1969). RECEIVED for review July 3, 1972 ACCEPTED March-12, 1973
Heat and Mass Transfer from Elliptical Cylinders in Steady Symmetric Flow Jacob H. Masliyahl and Norman Epstein” Department of Chemical Engineering, University of British Columbia, Vancouver,B. C.,Canada
Heat (or mass) transfer from an elliptical cylinder with a ratio of minor to major axis of 0.2 has been investigated b y the numerical solution of the energy (or diffusion) equation for constant fluid properties in steady symmetric flow. The solution was effected in conjunction with the previously solved equations of continuity and motion, in elliptical coordinates. A Reynolds number range of 1-50 was covered for a Prandtl number of 0.7. Isotherms as well as local and surface-mean Nusselt numbers are presented. Friedlander’s approximate method was also used to compute surface-mean Nusselt numbers for a Prandtl number range of 0.7-50,000 at a Reynolds number of unity for the case of flow along the minor axis, and at Reynolds numbers of 1, 5, and 25 for the case of flow along the major axis, as well as for flow past a circular cylinder at Re = 1. The more exact and the approximate results are compared with each other and with experimental data from the literature.
Experimental and theoretical analysis of heat and mass transfer in laminar flow past circular cylinders has received much attention in the literature. One of the immediate applications for such studies is in the theory of hot-wire anemometers where the Reynolds numbers are usually quite low (Collis and Williams, 1959). Similar analysis for elliptical cylinders is scarce.
It is possible to treat both heat transfer and mass transfer from elliptical cylinders in a n analogous manner when the fluid properties are independent of pressure, temperature, and concentration, and when the concentration gradient is small and the energy dissipation LS neglected. Then, the energy equation in nondimensional form becomes
1 Present address, Department of Chemistry and Chemical Engineering, University of Saskatchewan, Saskatoon, Sask.
Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
317
where the dimensionless velocity vector, ;,is a strong function of Reynolds number, Re. The corresponding equation for mass transfer (diffusion equation) may be given by interchanging the normalized temperature, T , with the corresponding concentration and the Prandtl number, Pr, with the Schmidt number. The dimensionless velocity vector, is determined by the solution of the momentum and continuity equations. In contrast to axisymmetric bodies, for = 8 (pure conduction), there does not exist a solution for steady-state pure conduction from a heated elliptical cylinder (a circular cylinder and a flat plate being limiting cases of an elliptical cylinder) in an unbounded medium. For the case of low Reynolds numbers (Re < 1) and R e - P r ( = P e ) Re > Friedlander plotted the variation of N u / P ~ ’ /with ~ Re for Pr >> 1. The velocity field of Tomotika and Aoi (1950) for low Re was used. His results are in fair agreement with the experimental work of Davis (1924) and of Dobry and Finn (1956). Dennis, et al. (1968), obtained numerical solutions to the energy equation for the case of a circular cylinder for 0.01 Re 6 40 and tabulated the surface-mean Nusselt number for various values of Pr. Similarly, Dennis and Smith (1966) obtained the variation of Nu with Re and Pr by numerically solving the momentum and energy equations for the case of a flat plate parallel to the flow. X o theoretical work is available for the case of heat transfer from elliptical cylinders. I t is the intention of this paper to explore intermediate Reynolds numbers, 1 Re 6 50, for a Prandtl number of 0.7 (that of air). Friedlander’s technique is also employed to obtain N u for Pr up to lo4. An elliptical cylinder having an aspect ratio of 0.2 is studied with flow parallel to both the major and the minor axis, as well as a circular cylinder. For 1 -
> sin q and sinh [ = , the solution of eq 7 is given by Tomotika and Aoi (1953) and by Dennis, et al., as
T(t,q)
7
= e-X
5 A m K m ( x ) cos mq
m=O
(8)
where x = (Re.Pr.ee/8 sinh ta) and Am are functions of (Pe/sinh ta). Equation 8 satisfies the boundary conditions (6b) and (6c). Also, for large values of x (9) for all values of m, and eq 8 becomes
T([,q)
=
e-[x(cos
,,+l)+e/21H(qj
(10)
where H ( q ) is a function of q and is only known when A , is known. I t follows then, that for given (Pe/sinh Ea) and angle q and consequently for a fixed value of H ( q ) , one obtains
T(tb, 7)
=
T ( t c ,dlexp[(cos 7 [X(tb)
k
btl
(5) For the case of flow parallel to the minor axis, the dimensionless form of the energy eq 5 can be easily obtained by replacing each sinh by i cosh .$,cosh [ by i sinh 5, and c by - ic. Using this procedure, the characteristic length becomes 2c cosh Ea, which is the major axis of the elliptical cylinder. The velocity components ut and v,, for the steady symmetric flow past elliptical cylinders were obtained by the numerical solution of the Navier-Stokes equations together with the continuity equation, recently reported by the present authors (Masliyah and Epstein, 1971). The solution of the energy eq 5 is subject to the boundary conditions
a t the front stagnation point (7 = 0) of the elliptical cylinders is shown in Table I.
50-
20-
05
THEORETICAL FRIEDLANDER'S PPFRMCH. THIS WORK 0 DENNIS AND SMITH 0 FINITE DIFFERENCES, THIS WORK
-
EXPERIMENTAL
1 10
1
# , # > # < > I,
, , 1 1 # 1 1 1
ID
IO2
1
I
I
I , , , , ,
,
,,,,,,
103
04
PI
Figure 7. Variation of mean Nusselt number with Prandtl number for flow past a circular cylinder at Re* = 1
There appears to be a tendency for Xu*(0)/Re*'/2 to reach a constant value, in accordance with boundary layer theory. The variation of the surface-mean Kusselt number, Nu*, with Reynolds number, Re*, is shon-n in Figure 6 for Pr = 0.7, as computed by the finite difference method. For the case of a circular cylinder (O.R.= 1/0.995) at Re* = 1 and 5, the agreement of the computed data with the experimental results of Collis and Williams (Pr = 0.73) is good. At Re* = 1, the result for O.R. = 5 (flow along major axis) is practically coincident with that for O.R. = 0.2 (flow along minor axis), the computed Nusselt numbers being 1.076 and 1.082, respectively, indicating that a t this Peclet number, Pe* = 0.7, the flow direction is not critical. The values of -%* for O.R. = 5 are very close to those of a flat plate parallel to the Row (O.R. = a), given by Dennis and Smith, as would be expected. It is also interesting to note that, for a given elliptical cylinder with a ratio of minor to major axis of 1 : 5 , constant fluid properties, and a fixed undisturbed fluid velocity, C-, Figure 6 shows that the heat transfer coefficient is higher with the flow parallel t o the major axis than that with the f l o parallel ~ t o the minor axis. Such a gain in heat transfer is accompanied by a lower drag force, as shown in Table I by Masliyah and Epstein (1971). Heat transfer using elliptical cooling rods should therefore be investigated in systems, such as blood flow, where it is desirable to have maximum heat transfer with minimum pressure drop, due to sensitivity of the fluid to shear stress. At Re* = 1, Friedlander's approach to computing %*was extended down to Pr = 0.7 for the case of a circular cylinder, as shown in Figure 7. This approach gives reasonable agreement with the experimental results of Kramers (1946) and of Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
32 1
Table II. Values of L*/Pr’/a Calculated by Friedlander Approach Pr
Re* = 1
0.7 1 4 5 10 30 50 100 400 500 1,500 5 ,000 10,000 30 ,000 50 ,000
1.150 1I111
O.R. = 5 Re* = 5
...
...
1.794 1.770 ... 1.730 1.723 ... 1.710 1.705 1.702 1.702 ...
...
... ... ...
1.921 1.897
0.979 0.947 ... 0.898 0.884 0.869 0.864 0.861 0.861 ... ...
Re* = 25
... 3.63
... 3.62 3.61 .
...
LTlsamer (1932), as Jvell as with the numerical work of Dennis and Smith and with the present finite difference solution. The difference between the approximate and the more exact solutions is only about lo%, indicating the usefulness of Friedlander’s approach in giving fairly accurate results with relatively little computational time. The dependence of the surface-mean Kusselt number of the elliptical cylinders on Pr1l3up to very large values of Pr, and Re* up to 25, using Friedlander’s method is given in Table 11. The term T U * / P ~ ~approaches ’~ a constant value as P r increases, in accordance with the theoretical work of Lighthill (1950).
.
I
3.60 3.60 3.60 ...
322
... 0.852 0.782
... 0.721 0.672
...
... ...
0.616 0.607 0.594
...
0.590
Pr Re Re*
T
T’ T,’
u
U V’
= Prandtl number, cp p / k
Reynolds number = 2apU/p for O.R. 6 1 and = 2bpU/p for O.R. 2 1 = Reynolds number = 2upU/p for all values of O.R. = dimensionless local temperature = (T’ T-’)/(T0’ - T-’) = dimensional local temperature of fluid = dimensional temperature of fluid at infinity = dimensional velocity of the undisturbed fluid at infinity = dimensionless local velocity = v’/U = dimensional local velocity of fluid =
- , /
\
~
Euler’s constant (= 0.5772) elliptical coordinate, angle n-9
viscosity of fluid elliptical coordinate normal t o q density of fluid dimensionless stream function = +’/uL’ for O.R. 6 1 and = +‘/bU for O.R. 1 dimensional stream function nondimensional del operator nondimensional Laplace operator
>
cylinder surface outer thermal envelope one grid length in E away from the thermal outer envelope toward the cylinder surface edge of the thermal boundary layer value a t the j t h grid in E q direction direction
Gi
Pe Pe *
1.112 1.071 0.923
0.633 0.632 0.632
major semi-axis of elliptical cylinder - length of minor semi-axis of elliptical cylinder focal length of the elliptical coordinate system thermal capacity of fluid coefficient in eq 18 value of coefficient C a t 5 = coefficients of regression eq 19 dimensionless circumference = Cr’/c circumference of elliptical cylinder coefficient in eq 18 value of coefficient D a t 5 = coefficients of regression eq 20 total amount of diffusing heat, nondimensionalized (see Friedlander, 1957) local heat transfer coefficient function of 7, [related to cos rnv and Pe/ sinh €a
O.R.
0.863 0.832 0.756 ... 0.719 0.685 ... 0.662 0.646 ...
...
- length of
b
O.R. = 0.2 Re* -- 1
...
Nomenclature a
O.R. = 1 Re* = 1
-
thermal conductivity of fluid modified Bessel function local Nusselt number = 2ah/k for O.R. 6 1 and = 2bh/k for O.R. >/ 1 local Kusselt number = 2ah/k for all values of O.R. surface-mean Nusselt number = 2ah/k for O.R. 6 1 and = 2bh/k for O.R. >/ 1 surface-mean Nusselt number = 2ah/k for all values of O.R. orientation ratio = ratio of the axis parallel to the net flow direction to that perpendicular t o the flow Peclet number = Re.Pr Peclet number = Re*.Pr
Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
a dimensional quantity surface-mean value vector literature Cited
Cole, J., Roshko, A., “Proceedings of the Heat Transfer and Fluid Mechanics Institute,’’ p 13, Stanford University Press, Stanford California, 1954. Collis, D. d., Williams, 31.J., J . Fluid Mech. 6 , 357 (1959). Davis, A. H., Phil. Mag. 47, 1057 (1924). Dennis, S. C. R., Hudson, J. D., Smith, N., Phys. Fluids 11, 933 (1968).
Dennis, S. C. R., Smith, N., J . Fluid Mech. 24,509 (1966). Dobry, R., Finn, R. K., I n d . Eng. Chem. 48, 1540 (1956). Friedlander, S. K., A.Z.Ch.E. J . 3 , 4 3 (1957). Hieber, C. A., Gebhart, B., J . Fluad Mech. 32, 21 (1968).
Kaplun, S., J. Math. Mech. 6, 595 (1957). Kramers, H., Physica 12, 61 (1946). Lighthill, AT. J., Proc. Roy. Soc., Ser. A 202, 359 (1950). Masliyah, J. H., Znt. J. Heat Mass Trans er 14, 2164 (1971). Masliyah, J. H., Epstein, N., IND. ENG. HEM., FUNDAM. 10, 293 ( 197 1). Masliyah, J. H., Epstein, N., Progr. Heat Mass Transfer 6, 613
d
I1 ,- Q72). - . - ,.
Oseen C. W., “Hydrodynamik,” Lei zig, 1927. Taneda, S., Rep. Res. Znst. Appl. d c h . , Kyushu Univ. (1968).
16,
155
Taneda, S., Honji, J., J. Phys. SOC.Jap. 30, 262 (1971). Tomotika, S., Aoi, T., Quart. J . Mech. Appl. Math. 3 , 140 (1950). Tomotika, S., Aoi, T., Quart. J . Mech. Appl. Math. 6, 290 (1953). Ulsamer J., F O T S CGeb. ~ . Zngenieurw. 3, 94 (1932). Wood, d’. W., J . Fluid Mech. 32, 9 (1968). RECEIVED for review July 17, 1972 ACCEPTEDMarch 8, 1973 The authors are indebted to the University of British Columbia and to the National Research Council of Canada for financial support.
State-Space Formulation of Fixed-Bed Reactor Dynamics M. 1. Michelsen,’
H. 8. Vakil,
and A.
S. Fobs”
Department of Chemical Engineering, University of California, Berkeley, Calif. 94720
Locally linearized partial differential equations describing plug-flow reactor models are converted to a set of coupled ordinary differential equations using orthogonal collocation in distance. The resulting equations are amenable either to a state-space form or to a rational polynomial transfer function representation. An adiabatic fixed-bed reactor with a homogeneous exothermic reaction was studied. The steady-state and the dynamic behavior i s adequately represented b y a sixth-order model, the maximum error in the transfer functions being less than 4 X 1 O-2. Higher order models yield an improvement of about a factor of 5 for each additional collocation point. The success of the model i s furthermore reflected b y the accurate placement of the dominant zeros of the reactor transfer functions.
Locally linear plug-flow models in the form of partial differential equations have been shown to represent well the principal dynamic behavior of fixed bed reactors (Hoiberg, et al., 1971; Sinai and FOSS,1970). The direct use of such models, however, in control analyses and other applications is severely hampered by analytical and computational complexity that is directly attributable to the partial differential equation form of the model itself. A simplified lumped approximation incorporating accurately the essential features of the model is therefore highly desirable. Such a simplification is obtained here by the method of orthogonal collocation. I n this method the state variables are represented as a weighted sum of the states a t selected points in the reactor, the points in this case being the roots of the N t h degree shifted Legendre polynomial. It is shown here that N may be taken small, say 6 or 8, with excellent representation of the dynamic behavior. Such “efficiency” of approximation contrasts sharply with that of the usual “stirred-tank” approximation where as many as 1000 spatial subdivisions are necessary to achieve comparable accuracy. From the collocation approximation one may easily derive low-order transfer functions and ordinary differential equations in state-space form, both of which are given here for one particular type of reactor. The transfer functions are found useful for investigation of the accuracy of the approximation, primarily through the placement of the dominant On sabbatical leave from Instituttet for Kemiteknik, Danmarks Tekniske Herjskole, Lyngby, Denmark.
zeros. The state-space form of the approximation, on the other hand, is found to be of considerable value in analyses of reactor control systems (Vakil, et al., 1973). Reactor Equations and Collocation Approximation
The particular reactor to be investigated is a fixed-bed adiabatic reactor in which an exothermic noncatalytic reaction occurs in the fluid phase. The dimensionless equations describing the dynamics of such a reactor are (Crider and FOSS, 1968;Stangeland and FOSS, 1970)
bC
dC
bz
bt
-+-
=
-R(C,T)
dT + dz at
bT
-
=
= -kC
R(C,T)
exp
(T + T,) EoT
(la)
+ h(Tp- T )
(1b)
~
b_ Tp - hb(T - T P ) bt The reaction is assumed to be first order and irreversible with a rate expression of the Arrhenius type. Radial gradients and axial diffusion are assumed to be unimportant. Furthermore, intraparticle resistance to heat conduction and the wall heat capacity are neglected. At steady state, eq 1 with boundary conditions
C = l ; T=O
(z=O)
Ind. Eng. Chem. Fundam., Vol. 12, No. 3, 1973
(2) 323
