NATURAL CONVECTION IN HORIZONTAL CONCENTRIC CYLINDRICAL ANNULI R O B E R T L E M L I C H
L L O Y D CRAWFORD‘ AND
Department of Chemical Engineering, C’niuersity of Cincinnati, Cincinnati 21, Ohio
The problem of steady, laminar natural convection between horizontal concentric circular cylinders was attacked numerically via detailed difference equation approximations for the differential equations of conservation. No terms were eliminated by boundary layer arguments. The problem was solved with machine language on an IBM 650 digital computer for several diameter ratios and several Grashof numbers at a Prandtl number of 0.714. Computed results agree well with experiments reported in the literature. In the appropriate limiting case, the computed results also accord with an analytical derivation for creeping flow.
N
CONTRAST
with the situation for the flat plate or the un-
I bounded cylinder, theoretical study of natural convection
between horizontal concentric cylinders has received virtually no attention in the literature. This, no doubt, is due a t least in part to the relatively more complicated nature of the system. Recent experimental studies with concentric cylinders (8) have verified that various conditions can produce widely varying flow patterns. Accordingly, the present article presents the results of a theoretical investigation of the problem. The problem was attacked by setting u p the “complete” steady-state momentum, energy, and continuity equations, introducing the stream function, rewriting the resulting simultaneous differential equations as finite difference equations, and solving them on an IBM 650 computer with machine language. None of the usual boundary layer assumptions were required, so that the results presented here are free of such restrictions. For natural convection in general, there are very few such numerical solutions in the literature, even with boundary layer assumptions. Recently, Hellums and Churchill ( 6 ) studied the vertical flat plate in natural convection, using a different approach from that of the present authors, and Bodoia and Osterle (3) published a study of natural convection between a pair of vertical plates. Analysis
The analysis began with the well known (7) fundamental two-dimensional, differential conservation equations of mass, momentum, and energy, for constant fluid properties (except buoyancy) and negligible frictional and compressional heating. (For negligible pressure changes, the buoyancy is a function of temperature only.) By introducing the stream function $ and an arbitrary reference velocity V , these equations were combined and, after some manipulation, simplified to yield dimensionless Equations 1 and 2 in polar form:
Present address, E. I. du Pont de Nemours Br Co., Inc., Buffalo, N. Y. 260
I&EC FUNDAMENTALS
Dimensionless velocity u ’ = u / V , so that in Equation 1 the dimensionless stream function $’ = $/ab‘. The stream function itself is defined in the usual way for polar coordinates, namely uo =
-a&/&, so that by continuity, u,
1
w
= --
7.68’ The use of reference velocity V is in no way restrictive and has no physical significance here. It is employed merely for convenience. Reference Reynolds number, Re, which is defined in terms of V, is employed as a numerical scale factor rather than a parameter of the system. The dimensionless temperature and radius are defined by: T‘
=
~
T - TR Ti - TR
Operator
and r’
=
r/a
-D,Dt ’ = pa oDt
and for steady-state. The boundary conditions are no slip and uniform temperature at the bounding circumferences. Thus. at the inner radiusr’ = 1, u,’ = us’ = 0, and T’ = 1. At the outer radius, r’ = R, = uo’ = 0. and T’ = 0. By symmetry, ug = 0 along the vertical axis, so the stream function is conveniently taken as zero along this axis. Therefore, at r’ = 1 and r’ = R, I$‘= B+’,Qr’ = 0. Basically, the conversion from differential to difference equations follou ed conventional .’central difference” procedures ( 5 ) . In particular, here x is either r’ or 8. and f is either $ ’ o r T‘: ~1,’
+
where j j means f ( x j A x ) , and the functional values f are equally spaced. Equations 3a to 3d were then substituted into Equations 1 and E . Substitutions for the mixed derivatives were obtained by the appropriate combinations of the above. Actually, the individual Equations 3a to 3d are just first approximations to their respective derivatives. They are really the leading ternis of series of central differences of increasing orders. Accordingly, in regions of sharp change in T‘ and $’? additional terms were used to improve the approximation. Similarly, for the largest diameter ratio of 57, a finer grid was used in the vicinity of the inner cylinder and the wake then was employed elsewhere in the field. Ultimately, the over-all fineness was limited by the memory storage of the computing machine. Table I gives the range of variables and a quantitative description of the grid. The Prandtl number was 0.714 for ail cases. Lagrangian interpolation was used to provide “fictitious” values of R’ and 4’ in the region of changeover from coarse grid to fine. Special treatment of the stream function near the inner and outer bounding circumferences was necessary because the difference equations required values beyond the boundaries. These values were obtained by extrapolation combined with the boundary restriction d$’/dr’ = 0. At each grid point, the two algebraic equations for $’ and T’ were rearranged to express these quantities in terms of $’ and T’ a t neighboring and more distant points as required by the level of the approximation. T h e equations were lengthy and nonlinear. However, central $’ was itself linear, and no difficulty was presented in calculating the residual. T h e residuals !\’ere brought to virtual zero by a stepwise Gauss-Seidel (9)process, moving from point to point in a n orderly manner, alway:; using the most recent values of surrounding +’ and T‘ in calculating $’ and T’ a t a given point. This procedure of successive approximations was used rather than that of always seeking and then relaxing the largest residual to zero. The latter procedure, although convenient for certain types of hand calculation, was relatively impracticable to program for the computer. I n actual practice, intermediate results were occasionally read out of the computer, the trend of change \\’a:; noted, and then manually controlled jumps were incorporated to speed convergence of the solution. Some instability \vas encountered, especially with the largest diameter ratio of 57. This difficulty was met by deliberate damping (equivalent to under-relaxation). Instead of forcing the residual all the way to zero each time, it was reduced only fractionally. This, of course, required more steps before virtual zero was reached.
Results
Results of the computation are summarized in Figure 1. For the diameter ratio of 57, detailed magnifications in the vicinity of the inner circumference have been filed elsewhere (4,along with further details of analysis and results. The figures show the profiles for the dimensionless temperature and stream function. From the latter, the local velocity at any point can be determined. The local Nusselt number can be calculated from Nu = 2 dT’/dr’ a t the surface. The plots for $’ show that the fluid circulates in a characteristic twin kidney-shaped pattern, with the centers of rotation located above the horizontal axis. The velocity is low near the centers of rotation and the bottom of the annulus. These results are in accord with the appropriate qualitative flow patterns obtained in the recent experimental study by Liu, Mueller, and Landis (8). These authors reported oscillations as G r increases, especially for large diameter ratios. The approach to such conditions may well be related to the previously mentioned instability encountered by the present authors in their solution for their largest diameter ratio a t maximum Gr. The temperature profiles in Figure 1 show a generally upward distortion from that which would exist for pure conduction. This, of course, is to be expected and results from the rising currents of fluid. These results are also in qualitative accord with those of the aforementioned experimental study. Unfortunately, quantitative comparison is impractical. G r and the diameter ratios for the two studies do not coincide. Furthermore, the experimental investigators have indicated that the thermocouple used in developing their profiles for air was subject to indeterminate radiation error. Inspection of the figures reveals that the distortion in temperature profile (from that for pure conduction) increases with diameter ratio and with Gr. This latter is especially evident from a comparison of Figure 1, B and C. Figure 1 B shows relatively little distortion in T’. This means the conditions therein approach creeping flow ; that is, the convective terms in the differential equations are negligible. Accordingly, it should be possible to make an approximate comparison bebveen the computed $‘ values with those from a creeping flow solution. For this purpose, such a solution was derived analytically. The results of this derivation are presented below. Details of the derivations are available ( 4 ) .
M is the following determinant: Table I.
M = 4 In R (RZ
Range of Variables and Description of Grid
- R-2)
+ 8 - 4 (I?* +
(5)
R-2)
Pr == 0.714 for all cases
Diameter Ratio 2 8
Gr
A0
AY‘
7r
100,000
12
0.05
8 and 100
12
0.5
7r
Diameter ratio = 8, Grashof number = 8
Dtyeerence,
$’
1, 2, and 4
8 and 32
Table II. Comparison of Dimensionless Stream Function from Computer Solution with That from Analytical Creeping Flow Solution
{
0.5forr’ofl to5 2 , 0 for of 5 to 17 4 . 0 for Y ‘ of 17 to 57
r‘ 2 3 4
5 6 7
Creeping Jam 0.242 0,531 0.650 0.574 0.360 0.119
VOL. 1 NO. 4
Computer 0.290 0.579 0.685 0,594 0.369 0.121
% 18.1 8.7 5.2 3.4 2.5 1.7
NOVEMBER 1 9 6 2
261
Figure 1 . Computed values of
+’and J’
Diameter
Ratio
Gr
A
2
100,000
B
8
8 100
E
8 57 57 57 57 57
F G
H
262
l&EC FUNDAMENTALS
1
2
4 8 32
I
DIAMETER RATIO
SYMBOL
I
1.2
I
SYMBOL
DIAMETER RATIO
I
’
I
I
0.0-
-
9 11
’ w
0.6-
0.6-
0.2 0.4
Figure 2. Comparison Mueller, and Landis (8)
The coefficients
ca =
61, c2, 6 3 , c4
are as follows:
= -R4
=
Table I1 shows a comparison of $’ from the computer solution (Figure 1B) us. $’ from the creeping flow solution along the horizontal axis with diameter ratio of 8 and G r of 8. Of course. perfect agreement could not be expected since the conditions for Figure 122 only approach creeping flow. Severtheless, except near the inner cylinder, the agreement is reasonably good. This lends further support to the validity of the computer solution. Also, the nearly coiicentric, circular temperature profile of Figure 1B very nearly follows the logarithmic positioning of creeping flow (or of conduction). This in itself helps to show that the numerical solution correctly approaches the limiting situation of creeping flow and conduction, thus lending still further support to the propriety of the general numerical procedure employed. Table I11 shows the ,average Susselt numbers computed for the eight cases studied. Since h is based on over-all temperature difference, Nu for the inside diameter equals A-u for the outside diameter.
-
Table 111.
Diameter Ratio 2 8 8 57 57 57 57 57
Gr
Figure 3. Comparison of results with the correlation of Beckmann (2)
R4 - R2 - 1
c4
LOG
the correlation of Liw
-t 2R2 - 1 + 4R2 (In R)2 (6) + R‘-a - 2 In R ( 4 R2 In R + 2 - R2 - R-*) (7) -R2 4-2 - R - 2 + 21n R ( 2 R 2 h R + 2 - R2 - F 2 ) ( 8 ) 2 ( - R 4 4- R2 + 1 + 8 In R ( R 2 - 1) ( 9 ) c1
c: =
of results with
Average Nusselt Numbers
G7 100,000 8 100 I
2 4 8 32
NU 5.C88 0.962 1.566 0.847 0.948 1.061 1.183 1.434
Figure 2 shows a comparison of the present results with the correlation of Liu and others ( 8 )which is based on experimental data for air and liquids. [The Grashof number in these correlations is based on the gap width rather than the inner diameter. The ordinate k , / k = (Nu In R ) ] . The agreement is fair and approximately within the scatter of the experimental data on Ivhich the correlation is based. In viewing this comparison, the wide difference between the independent variables of the two investigations should also be considered. I n the work of Liu and others the diameter ratio did not exceed 7.5, while most of the present study was for a ratio of 57. Furthermore, their Gr for air was generally much higher than G r employed here. Better agreement is obtained when comparison is made against the classical experiments of Beckmann (2, 7) which were carried out with air and hydrogen. This is shown in Figure 3. (The upper two curves were prepared by small interpolations between neighboring curves of Beckmann’s own plot, while the lowest is one of his original curves). This figure reveals good agreement between the computed results of the present study and the older experimental results. Conclusions
By approximating the steady-state differential equations of conservation with appropriate difference equations, a system in steady natural convection can be solved numerically on a digital computer, even on one of “medium” size. No terms in the equations need be eliminated by boundary layer arguments. For the problem of horizontal concentric cylinders, which was that selected here, the computed results reveal characteristic twin kidney-shaped circulation patterns. Computed results agree well with experiment. They also agree in the limiting case with the analytical solution for creeping flow. A faster machine with a greater memory is desirable in problems of this sort, to permit use of a finer grid. Nomenclature a
A
= =
radius of inner cylinder radius of outer cylinder VOL. 1
NO. 4
NOVEMBER 1 9 6 2
263
heat capacity a t constant pressure diameter of inner cylinder, 2a differential operator Grashof number, D3p2gSp(T I - T R ) / p 2 dependent variable acceleration of gravity local coefficient of heat transfer at inner circumference general integer thermal conductivity determinant Nusselt number, h D / k Prandtl number, c , p / k polar coordinate (radial distance) radius (or diameter) ratio, A / . reference Reynolds number, D V p / p time temperature temperature a t inner surface temperature a t outer surface velocity component reference velocity independent variable volumetric coefficient of expansion polar coordinate (position angle measured from the positive horizontal axis) dynamic viscosity density stream function partial differential operator partial differential operator
G;.
f g
h
j
k
M NU
Pr r
R Re t T Ti
TR U
V x
B
e
P
$
V2 v4
Subscripts c
i
=
equivalent for convection
= general integer
- a ) replacing D a particular grid point the radial direction the outer surface the tangential direction
L
= with quantity ( A
0
=
r
R 9
at = in = at = in
Super scripts
’ = dimensionless quantity - = average Acknowledgment
The authors are grateful for the extensive use of the facilities at the University of Cincinnati Computing Center. literature Cited (1) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport
Phenomena,” IYiley, New York, 1960. (2) Beckmann, W., Forsch. Gebiete Ingenenieurw. 2, 165 (1931). )3) Bodoia, J. R., Osterle, J. F., J . Heat Transfer 84, 40 (1962). 4) Crawford, L. W., Ph.D. thesis, University of Cincinnati, 1961. (5) Hartree, D. R., “Numerical Analysis,” p. 55, Oxford at Clarendon Press, 1952. (6) Hellums, J. D., Churchill, S. W., International Developments in Heat Transfer, Paper 118, Boulder, Colo., .4ugust 1961. (7) Jakob, M., “Heat Transfer,” Vol. 1, p. 541, Wiley, New York, 1949. (8) Liu, C. Y.: Mueller, I\’. K., Landis, F., International Developments in Heat Transfer, Paper 117, Boulder, Colo., August 1961. (9) v l n e , LV. E., “Numerical Solution of Differential Equations,” Wiley, New York, 1953. RECEIVED for review November 27, 1961 ACCEPTED August 20, 1962 Work supported by National Science Foundation under research grant G-3060.
INERTIAL IMPACTION ON SINGLE ELEM ENTS M. N. GOLOVIN A N D A. A.
P U T N A M
Battelle Memorial Institute, Columbus, Ohio
United States and foreign open literature and various unclassified technical reports on the inertial impaction of small aerosol particles are reviewed. The data presented herein have possible practical application to the design of dust filters, heat exchangers, and combustion devices. The information surveyed is presented in terms of an inertial impaction efficiency E as a function of an inertial impaction parameter K, as originally defined b y Langmuir. A third parameter p is used to indicate deviations from Stokes’ law of resistance a t high particle Reynolds numbers. All data presented pertain to single elements such as cylinders, spheres, plates, and airfoils, no attempt being made to describe behavior when these elements are The theoretical treatments are based on the assumption of potential flow in the
in cascade arrangement. main body
of fluid.
DEPOSITION by the inertial impaction mechanism is of great interest to industry and science. This mechanism applies to many processes in nature as well as in man-made devices. Typical examples are the icing of airplane wings and control surfaces, the deposition of dust in certain filters, the operation of cascade impactors for aerosol sampling, the after-burning of dusty industrial exhaust gases, the recovery of products from a dispersed state, and the buildup of deposits in heat exchangers, furnaces, and aircraft propulsion systems utilizing high-energy fuels. Because of its wide applicability and because of the relative simplicity of mathematical formulation, the inertial impaction concept has received much attention.
A
264
EROSOL PARTICLE
ILEC F U N D A M E N T A L S
The present study concerns that range of particle sizes and other variables in which inertial impaction is the dominant mode of deposition. Other possible mechanisms of deposition, such as the interception effect observed in fiber air filters and the variously caused mechanisms of diffusional deposition which normally affect particles in the submicron range, are not considered in detail in this discussion. [The interception effect normally becomes important only when the ratio of the diameter of the particle to the diameter of the collecting shape (target) is roughly 1 to 10 or greater. However, when both impaction deposition and diffusional deposition are very small, interception effect can be the only deposition mechanism at much smaller size ratios.] When rarefied atmosphere, high
