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Flow and heat transfer in a pipe containing a coaxially-rotating disk
Fluid Dynamics Research 26 (2000) 377–391
Flow and heat transfer in a pipe containing a coaxially-rotating disk Janusz Wojtkowiak a , Jae Min Hyunb;∗ a
b
Institute of Environmental Engineering, PoznaÃn University of Technology, Piotrowo 3a, 60-965 PoznaÃn, Poland Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-ku, Taejon 305-701, South Korea Received 6 May 1999; received in revised form 2 July 1999; accepted 27 August 1999
Abstract A numerical investigation was made of steady laminar ow and convective heat transfer in a pipe constricted by a coaxially rotating disk. The analysis was executed by using the nite volume approach. Calculations were made for the through- ow Reynolds numbers (based on the pipe radius) of 10, 25, 50, 75, 100, 125, 150, the rotational Reynolds numbers of 0, 250, 500 and 1000, for the disk-to-pipe radius ratios of 0.9, 0.95 and 0.99, and for the Prandtl numbers of 0.7 and 7. The heat transfer rate, the pressure drop coecient, and the temperature distributions are determined. The results show that the temperature and ow characteristics are substantially a ected by the rotation of the disk. In the disk downstream- ow, the wall and disk recirculation zones are noticed. The swirl imparted to the ow measurably increases the heat transfer rate. The increase is especially noticeable when the wall recirculation region does not exist. Physical rationalizations are made of the computed ow features, and a brief description is given of the Nusselt number distribution. c 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
Keywords: High blockage ow; Heat transfer augmentation; Numerical simulation
1. Introduction Laminar ow of an incompressible viscous uid through a narrow annular passage in a pipe constitutes an essential building block of ow measurement technology and heat transfer enhancement techniques. The ow characteristics inside oating-element owmeters (rotameters), with a stationary or a rotating oat, were investigated numerically and experimentally by Buckle et al. (1992, 1995). ∗
Corresponding author. Tel.: +82-42-8693012; fax: +82-42-869-3210. E-mail addresses: [email protected] (J. Wojtkowiak), [email protected] (J.M. Hyun) c 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/00/$ 20.00 All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 9 9 ) 0 0 0 3 6 - 2
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Nomenclature A f l1 ; L1 l1 ; L2 ld ; Ld Min Nu Nuav p; P p; P Pr Q r; R rd ; Rd rp ; Rp Re Re! s t; T Tin t b ; Tb Tw u; U v; V w; W x y z; Z
uid thermal di usivity local pressure drop coecient distance between the inlet of the pipe and the disk upstream surface, ll = Ll =Rp distance between the downstream surface of the disk and the pipe outlet, l2 = L2 =Rp disk thickness, ld = Ld =Rp mean axial velocity of the uid, =Q=(R2p ) local Nusselt number average Nusselt number pressure, p = P=(Min2 ) pressure di erence between the inlet and outlet of the pipe, p = P=(Min2 ) Prandtl number, =vf =A volume ow rate radial coordinates, r = R=Rp disk radius, rd = Rd =Rp pipe radius, rp = Rp =Rp = 1 through- ow Reynolds number, =Min Rp =vf rotational Reynolds number, =!R2p =vf axial location of the wall separation point temperature, t = T=Tin temperature of the uid at the pipe inlet
uid bulk temperature, t b = Tb =Tin pipe wall temperature velocity components in the axial direction, u = U=Min velocity components in the radial direction, v = V=Min velocity components in the azimuthal direction, w = W=Min length of the disk recirculation region, x = X=Rp length of the wall recirculation regions, y = Y=Rp axial coordinates, z = Z=Rp
Greek letters
uid density vf
uid kinematic viscosity ! angular velocity of the disk A new type of piston- owmeter was proposed by Peters and Kuralt (1995), in which the uid moves through the narrow gap between the lateral surface of the piston and the inner surface of the pipe wall. Numerical evaluations of the system geometries, ow patterns and pressure distributions inside these owmeters with stationary and rotating pistons were conducted by Wojtkowiak et al. (1996, 1997). Several methods of heat transfer augmentation in laminar pipe ow have been summarized in the literature (Bergles, 1969, 1973, Kalinin et al., 1990). One popular method is to employ various types of concentric insertions (Rowley and Patankar, 1984; Vilemas et al., 1987; Budov and
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Dimitrev, 1989; Agrawal and Sengupta, 1989; Wu-Shung Fu and Ching-Chi Tseng, 1994), which generally demonstrates increased convective heat transports since swirl is imparted to the ow. The quantitative results are, however, not entirely consistent, and the detailed ow characteristics warrant an in-depth scrutiny. In the present paper, as a simpli ed paradigmatic model, the laminar ow passing through a circular pipe, which is constricted by a co-axially rotating disk, is considered. Comprehensive numerical computations are performed by solving the full Navier–Stokes equations. The ranges of parameters pertinent to practical applications are identi ed, and the local pressure elds, velocity pro les and Nusselt number distributions are exhibited. The global ow patterns are illustrated, and the behavior of secondary ow bubbles is depicted. The heat transfer rate, pressure drop coecient and temperature distributions are described. Emphasis is placed on delineating the changes in ow properties, which are brought forth by the rotation of the disk. The ranges of computational parameters are chosen as follows: the main- ow Reynolds number (based on the pipe radius) 106Re6150; the rotational Reynolds number of the disk 06Re! 61000; the Prandtl number Pr = 0:7 and 7; and the disk=pipe radius ratio rd = 0:9; 0:95 and 0.99. The thrust is given to the depiction and rationalization of the major ow features. A brief description will be made of the behavior of Nusselt numbers on the pipe wall. 2. Mathematical model and solution procedure Steady, axisymmetric, incompressible, laminar ow of a Newtonian uid is considered. The thermophysical properties of the uid are taken to be constant. With these assumptions, using cylindrical coordinates (r; ; z), with corresponding velocity components (v; w; u), the governing equations, in non-dimensional form, are written as 1 @(rv) @u + = 0; r @r @z
(1)
@v w2 @p 1 @v =− + v +u − @r @z r @r Re
@2 v 1 @v @2 v v + 2− 2 + 2 @r r @r @z r
@2 u 1 @u @2 u + + 2 @r 2 r @r @z
@w @w vw 1 v +u + = @r @z r Re
@2 w 1 @w @2 w w + 2 − 2 + 2 @r r @r @z r
"
;
(2)
!
@p @u @u 1 v +u =− + @r @z @z Re
1 @ @t @t @t 1 v +u = r @r @z Re Pr r @r @r
!
;
(3) !
;
(4)
#
@2 t + 2 : @z
(5)
In the above, nondimensionalization has been implemented using the pipe radius Rp , the average inlet velocity Min , and the average inlet temperature di erence of the uid (Tin − Tw ) as the reference
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Fig. 1. Computational domain and boundary conditions.
length, velocity and temperature, respectively. Consequently, the dimensionless variables are de ned as R Z U V W P T − Tw r= ; z= ; u= ; v= ; w= ; p= ; t= ; Rp Rp Min Min Min Min2 Tin − Tw (6) R2p ! Rp Min vf Re = ; Re! = ; Pr = : vf vf A The actual ow con guration and the computational domain are outlined in Fig. 1. Considerations are given to the proper boundary conditions. At the pipe inlet, the ow is assumed to be hydraulically fully developed, with the average velocity equal to one, and with the radially uniform temperature (t = 1), i.e., u = 2(1 − r 2 );
v = w = 0;
t = 1;
at z = 0;
06r ¡ 1:
(7)
At the pipe outlet, the ow is fully developed, with zero gradients of ow variables, i.e., @u @v @w @t = = = = 0; at z = l1 + ld + l2 ; 06r ¡ 1: (8) @z @z @z @z On all the pipe wall, no-slip conditions for the velocities and a constant temperature are speci ed, i.e., u = v = w = t = 0;
at 06z6l1 + ld + l2 ; r = 1:
On the upstream and downstream surfaces of the disk, the conditions can be expressed as Re! u = v = t = 0; w = r at z = l1 ; 06r6rd and at z = l1 + ld ; 06r6rd : Re On the lateral surface of the disk, similar conditions are applied, Re! u = v = t = 0; w = rd ; at l1 6z6l1 + ld ; r = rd : Re Symmetry conditions are speci ed on the axis, i.e., @t @u @w = = v = 0; at r = 0: = @r @r @r
(9) (10)
(11)
(12)
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The discretized representations of the governing equations (1) – (5), together with the boundary conditions (7) – (12), were solved by utilizing the well documented nite-volume, iterative algorithm, SIMPLER, of Patankar (1980). Convective terms were discretized by using the reformulated QUICK scheme of Hayase et al. (1992). This provides improved physical consistency of the numerical solution than other methods (e.g., upwind or hybrid central=upwind). Moreover, the treatment of source terms by Hayase ensures the continuity of the ux through the boundary, which contributes to numerical stability. The central di erence scheme was used for the di usive terms. The upstream length l1 and downstream length l2 were selected to allow a sucient entry length near the inlet and to justify the zero-gradient condition at the outlet (see Fig. 1). A trial-and-error method was used to reach the values l1 and l2 . For most conditions, the upstream length l1 = 5, and the downstream length l2 = 55 were found to be adequate, except for the highest Reynolds numbers in the computations (Re = 150; Re! = 1000), for which a longer downstream length (l2 = 75) was adopted. This elongation was introduced in order to minimize the in uence of the outlet boundary condition and to assist in obtaining a converged solution. The thickness of the disk was taken to be ld = 0:005. Extremely large values (1030 ) of viscosity and thermal conductivity, along with zero source terms, were speci ed for all the points within the disk in order to simulate the no- ow and perfect conductor conditions there. A staggered grid was adopted and grid stretching was implemented. Extensive trial calculations were carried out to ascertain the grid-convergence characteristics of the results. These comprehensive sensitivity tests examined the impacts of the variations in the total number of cells, locations of concentrated cells and grid expansion ratios, among others. For the majority of the runs, the 40×160 grid in the r–z coordinates was found to be optimal in terms of overall computational eciency and accuracy. Only for the larger downstream length (l2 = 75), the 40 × 180 grid was deployed. The general grid arrangement was such that the grid points were concentrated close to the disk surface, the disk edge and the pipe wall. In the axial direction 20 cells were placed in the upstream region of the disk, and 3 cells for the disk and 137 (157 for l2 = 75) cells in the downstream region of the disk. In the radial direction 9,7 and 5 cells were placed in the gap for rd = 0:9; 0:95 and 0.99, respectively. In all the cases, the solution was declared to have converged when the global residuals of the dependent variables were less than 10−4 . In the course of analyzing the numerical results, it is advantageous to introduce the following physical quantities. The coecient of local pressure drop is f=
P − 32=Re(L1 + Ld + L2 )=2Rp 12 Min2 16 (l1 + ld + l2 ); = 2p − 1 2 Re Min 2
(13)
where, the overall pressure drop P = Pin − Pout , in which Pin and Pout denote the pressures at the pipe inlet and outlet, respectively. In Eq. (13), 32 L1 + Ld + L2 1 M 2 Re 2Rp 2 in indicates the pressure losses for Hagen–Poiseuille pipe ow. The local Nusselt number at the pipe wall is 2Rp @T 2 @t Nu = − =− ; Tb − Tw @R R=Rp t b @r r=1
(14)
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and the average Nusselt number is 1 Nuav = l1 + l d + l 2 is
Z
l1 +ld +l2
0
Nu d z:
(15)
In the above, the bulk temperature of the uid in the upstream and downstream sides of the disk 2 Tb = Min R2p
Z
Rp
0
UTR dR;
tb = 2
Z 0
1
utr dr;
(16)
and the bulk temperature in the gap region is 2 Tb; g = Min R2p
Z
Rp
Rd
UTR dR;
t b; g = 2
Z
1
Rd
utr dr:
(17)
3. Results and discussion Before the main computations were started, veri cation tests were executed. For the testing calculations, the 40 × 160 staggered and stretched grid was employed. First, a straight pipe of length ll + ld + l2 = 60, with no obstructions, was used. The results were highly consistent with the existing analytical and experimental data (e.g., Kays and Crawford 1993, White 1991). In terms of local and average Nusselt numbers, pressure drop coecient and entrance length, the discrepancies were smaller than 6%. Next, test runs were made to compute the local pressure drop coecient for a stationary disk in a pipe. For all the through- ow Reynolds numbers, the di erences between the computed results and the experimental data of Idelchik (1996) were less than 10% for rd = 0:9, 13% for rd = 0:95 and 20% for rd = 0:99. Finally, in order to assess the performance of the present calculations for ows with separation, the ow in rotameter was computed. The numerical results were in broad agreement with the published data of Buckle et al. (1992,1995). Plots of the numerically constructed meridional stream functions are exempli ed in Figs. 2 and 3. Fig. 2 shows the case of a stationary disk (Re! = 0, and rd = 0:95). As Re varies, the ow structure on the upstream-side of the disk is a ected in a minor way, whereas the ow on the downstream side undergoes conspicuous changes. For a small Reynolds number (see Fig. 2a for Re = 10), the ow patterns are qualitatively similar on both sides of the disk, i.e., no separation, and no recirculation zones are visible. Physically, for a very low Reynolds number, the inertial forces are insucient to induce separation, and a general pattern is akin to a creeping ow. As Re increases (see Fig. 2b for Re = 50), an annular wall-jet is formed, and one large recirculation region on the downstream face of the disk is visible. For larger Reynolds numbers (see Fig. 2c for Re = 100), a new recirculation region begins to form on the pipe wall due to the adverse pressure gradient caused by the sudden expansion. On the downstream side of the disk, the increasing pressure slows down the wall ow and pushes the uid to move backward. A qualitatively similar sequence of events is observed for rd = 0:9 and 0.99. As rd increases, the formation of the recirculation regions takes place earlier, i.e., the separation bubbles occur at lower Re values. Fig. 3 depicts a representative series of computational results for the case of a rotating disk (Re! =500, and rd =0:95). The rotating disk induces axial ows toward its upstream and downstream surfaces. The uid then moves radially outward on the surface of the disk, and it travels along the
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Fig. 2. Plots of the meridional stream function for the case rd = 0:95 and Re! = 0: (a) Re = 10; (b) Re = 50; (c) Re = 100.
Fig. 3. Plots of the meridional stream function for the case rd = 0:95 and Re! = 500: (a) Re = 10; (b) Re = 50; (c) Re = 100.
pipe wall, away from the disk. The interaction of this ow with the main through- ow produces a complex pattern, as exhibited in Fig. 3. At a small Reynolds number (see Fig. 3a for Re = 10), the existence of the recirculation regions is apparent on both the upstream and downstream sides of the disk. The upstream-side bubble is much smaller than the downstream side one. This is due to the fact that, on the upstream-side of the disk, the main through- ow near the pipe wall, and the pipe wall ow driven by the rotating disk are in the opposite direction. On the downstream side of the disk, the above two wall ows are in the same direction. At a slightly higher Re, the upstream-side recirculation bubble vanishes, and the global ow pattern is dominated by the main through- ow. As Re increases further (see Fig. 3b for Re = 50), a “two-celled” structure of the disk
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Fig. 4. Schematic of the principal ow elements.
recirculation region is visible in the meridional plane. A similar “two-celled” internal structure has been reported in swirling pipe ow undergoing vortex breakdown (see, e.g., Faler and Leibovitch, 1978; Escudier, 1988; Brown and Lopez, 1990, and more recently Keller, 1995). The present results display qualitatively similar patterns. However, no de nitive conclusions can be drawn at this point, since there are notable di erences in the ow geometry and boundary conditions. In the present problem, the main (outer) disk recirculation zone is due principally to the abrupt change in the
ow geometry. In the case of conventional vortex-breakdown bubble, the dynamic unbalance in the swirling uid motion is cited to be the dominant cause. As Re increases further (see Fig. 3c for Re=100), an elongated separation bubble is formed on the pipe wall, and a strong interaction between the disk and the wall recirculation regions is noticeable. The size of the wall bubble increases, and the length of the disk recirculation zone is reduced. Simultaneously, changes in the internal structure of the disk region are evident. The “two-celled” structure is preserved, but the internal bubble moves closer to the disk surface. The above-described sequence is qualitatively similar for other values of Re! and rd adopted in the study. The foregoing observations based on the numerically constructed visualizations are summarized to produce a quantitative portrayal of the two principal ow elements, i.e., the disk recirculation zone and the wall recirculation zone. Fig. 4 o ers a schema of the ow description; i.e., the length of the disk recirculation region (x), the length of the wall recirculation region (y), and the location of the wall separation point (s). For a stationary disk (Re! = 0), Figs. 5 and 6, respectively, show the x–Re and y–Re plots. When the gap clearance is extremely small, x decreases as Re increases. As stipulated earlier, when the wall recirculation region is formed, the disk recirculation zone shrinks in size. Also, the wall recirculation region grows relatively fast as Re or rd increases. However, in the parameter space in which no wall recirculation zone exists, x naturally increases with increasing Re, and the results in Fig. 5 re ect these qualitative features. Similarly, Fig. 6 illustrates that y increases as Re or rd increases. The mechanism of the formation of the wall recirculation zone is the presence of a positive (adverse) pressure gradient in the downstream side of the disk. As noted, the pressure gradient grows rapidly with Re and with rd , which is in line with the above physical explanation. Figs. 7 and 8 present the results for a rotating disk. As Re or rd increases, y increases, which is qualitatively akin to the case of a stationary disk (see Fig. 6). However, y decreases as Re! increases. A plausible physical argument is that, as the disk rotates, the azimuthal velocity component is developed. This causes a further reduction of pressure in the direction of main ow, which gives
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Fig. 5. Disk recirculation region length x vs. Reynolds number for a stationary disk (Re! = 0).
Fig. 6. Wall recirculation region length y vs. Reynolds number for a stationary disk (Re! = 0).
rise to a favorable (negative) pressure gradient. Also, it should be pointed out that the pressure drop in the gap contributed by the rotating disk is much smaller than the pressure drop caused by the through- ow in the narrow clearance. In accord with these physical considerations, y is reduced as Re! increases and as Re decreases. The qualitative trends detectable in Fig. 8 are compatible with the foregoing physical interpretations. The wall separation point moves closer to the disk as Re and rd increase and as Re! decreases. Physically, the adverse (positive) pressure gradient strengthens as Re and rd increase, which leads to a reduction in s. In contrast, as Re! increases, the favorable (negative) pressure gradient strengthens, which causes an increase in s. The plots in Figs. 5 and 8 indicate that s is generally smaller than x. This implies intense interactions between the separation process and the disk recirculation zone.
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Fig. 7. Wall recirculation region length y vs. Reynolds number for rd = 0:95.
Fig. 8. Wall separation point position s vs. Reynolds number.
In Fig. 9, the relationship between the local pressure drop coecient (f) (see Eq. (13)) and the Reynolds number (Re) is presented. The f–Re variation when Re is small and moderate is typical of laminar ow conditions, i.e., a steep decrease of f is noticeable as Re increases. The asymptotic value at large Re is characteristic of a turbulent ow, for which, as well known, the local pressure drop coecient is approximately independent of the Reynolds number. Fig. 9 also clearly demonstrates the obvious dependence of f upon rd and Re! . It is seen that f increases with rd and, to a considerably lower degree, with Re! . When Re and rd are large, the e ect of Re! on
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Fig. 9. Local pressure drop coecient f vs. Reynolds number: (a) rd = 0:9; (b) rd = 0:95; (c) rd = 0:99.
f is negligible. In such cases, the pressure drop in the main ow is much larger than the additional pressure decrease due to the disk rotation. Next, heat transport characteristics are mentioned. In Figs. 10 and 11, the axial pro les of local Nusselt number on the pipe wall as well as the variations of the uid bulk temperature are illustrated. The general shape of Nu-pro les remains largely unchanged. At the entrance (z = 0), a uniform temperature pro le is speci ed, which gives a large value of Nu. Afterwards the thermal boundary layer grows, and Nu decreases. As the uid reaches the disk region, the uid is accelerated. The
uid impinges on the cold pipe wall, which leads to a sudden and extremely localized increase in Nu. Downstream of this disk region, Nu generally drops o sharply to a very low value, which is caused by reduced uid velocities owing out to the pipe outlet. The distributions of the Nusselt numbers downstream of the disk are inspected. The distributions re ect the complex ow structure in the downstream region of the disk. In general, when swirl is imparted to the ow the Nusselt number increases (compare e.g. Figs. 10a with 10b for z ¿ 5). However, when the wall recirculation region begins to form, some decrease in Nu is visible (see Figs. 10b and 11b for Re = 100). This is due to the fact that the wall recirculation zone causes the main through- ow to move away from
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Fig. 10. Bulk temperature t and local Nusselt number Nu for rd = 0:95 and Pr = 7: (a) Re! = 0; (b) Re! = 500.
the pipe wall. It is worthwhile to note that the minimum value of the Nusselt number occurs in the vicinity of the separation point of the wall recirculation region. For downstream close to the exit, Nu tends to settle down to the value for a fully developed pipe ow. The above numerical data indicate that the e ects of the Re and especially of Re! on Nu, in the downstream region of the disk, are substantial. Obviously, as Re and Re! increase, convective activities become more vigorous, which increases Nu. The pro les of the bulk temperature t are also included in Figs. 10 and 11. A monotonic decrease in t is seen in the upstream region of the disk. In the disk gap region, a rapid decrease in t is visible. In the downstream region, a mild decrease in t is evident. In general, the rate of decrease in t becomes more pronounced as Re decreases, and Re! increases. Minor irregularities in the bulk temperature plots, which are noticeable on the disk upstream-side (e.g., Fig. 10b for Re = 10), and on the disk downstream-side (e.g., Fig. 10a for Re = 100), are attributable to the existence of the upstream side (Fig. 3a), and the downstream side (Fig. 3c) wall recirculation regions.
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Fig. 11. Same as in Fig. 10, except for Pr = 0:7.
A combination of the reverse ow in these regions with the main ow produces the bulk temperature peaks, as exhibited in Figs. 10 and 11. For instance, in the exemplary plots of Figs. 10 and 11, the t b -curves for Re = 50 very close to the disk are strongly a ected by the relative strengths of upstream- and downstream-side recirculation regions. As remarked previously, the existence and size of these recirculation regions are extremely sensitive to Re and other ow parameters. In view of the complexity of the problem formulation, parametric evaluations of t b -distributions with Re, Re! and rd are beyond the scope of the present endeavor. A precise examination of heat transfer characteristics can be made by deploying far more grid points, and this task will be tackled in the forthcoming papers. Finally, the average Nusselt number Nuav is plotted in Fig. 12. Obviously, Nuav increases with Re, Re! and rd . It is worthwhile to note that the increase in average Nusselt number due to the decrease of disk=pipe clearance (from rd = 0:9 to rd = 0:95) is relatively small, whereas the increase in Nuav , due to the disk rotation, is substantial.
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Fig. 12. Average Nusselt number Nu av vs. Reynolds number for Pr = 7: (a) rd = 0:9; (b) rd = 0:95.
4. Concluding remarks Numerical results have been obtained for steady laminar uid ow and heat transfer in a pipe with a coaxially rotating disk. It is shown that the rotation of the disk has profound impact on the
ow and heat transfer characteristics especially in the downstream side of the disk. A large value of disk=pipe radius ratio causes a high-pressure loss and leads to the formation of wall recirculation regions. These contribute to the lowering of Nu. The swirl imparted to the ow augments both heat transport and pressure loss, but the increase in pressure loss is limited.
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Acknowledgements Appreciation is extended to the referee whose constructive comments led to improvements in the revised manuscript. References Agrawal, A.K., Sengupta, S., 1989. Laminar ow and heat transfer in blocked annuli. Num. Heat Transfer (Part A) 15, 489–508. Bergles, A.E., 1969. Survey and evaluation of techniques to augment convective heat and mass transfer. In Progress in Heat and Mass Transfer. Pergamon Press, Oxford, Vol. 1, pp. 331– 424. Bergles, A.E., 1973. Recent developments in convective heat transfer augmentation. Appl. Mech. Rev. 26, 675– 682. Brown, G.L., Lopez, J.M., 1990. Axisymmetric vortex breakdown. Part 2. physical mechanisms. J. Fluid Mech. 221, 553–576. Budov, V.M., Dimitrev, S.M., 1989. Compact Heat Exchangers of Nuclear Power Stations. Energoatomizdat, Moskva (in Russian). Buckle, U., Durst, F., Howe, B., Melling, A., 1992. Investigation of a oating element owmeter. Flow Meas. Instrum. 3, 215–225. Buckle, U., Durst, F., Kochner, H., Melling, A., 1995. Further investigation of a oating element owmeter. Flow Meas. Instrum. 6, 75–78. Escudier, M., 1988. Vortex breakdown: observations and explanations. Prog. Aerospace Sci. 25, 189–229. Faler, J.H., Leibovitch, S., 1978. An experimental map of an internal structure of a vortex breakdown. J. Fluid Mech. 86, 313–335. Hayase, T., Humphrey, J.A.C., Greif, R.A., 1992. Consistently formulated QUICK scheme for fast and stable convergence using nite-volume iterative calculation procedures. J. Comput. Phys. 98, 108–118. Idelchik, A.V., 1996. Handbook of Hydraulic Resistance, 3rd Edition. Begell House, New York. Kalinin, E.K., Dreicer, G.A., Jarho, S.A., 1990. Heat Transfer Augmentation in Channels, 3rd Edition. Masinostroenie, Moskva (in Russian). Kays, W.M., Crawford, M.E., 1993. Convective Heat and Mass Transfer, 3rd Edition. McGraw-Hill, New York. Keller, J.J., 1995. On the interpretation of vortex breakdown. Phys. Fluids 7, 1695 –1702. Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York. Peters, F., Kuralt, T., 1995. A gas owmeter of high linearity. Flow Meas. Instrum. 6, 29–32. Rowley, G.J., Patankar, S.V., 1984. Analysis of laminar ow and heat transfer in tubs with internal circumferential ns. Int. J. Heat Mass Transfer 27, 553–560. Vilemas, J., Cesna, B., Survila, V., 1987. In: Zukauskas, A., Karni, J. (Eds.), Heat Transfer in Gas-Cooled Annular Channels. Hemisphere, Washington, DC. White, F.M., 1991. Viscous Fluid Flow, 2nd Edition. McGraw-Hill, New York. Wojtkowiak, J., Kim, W.N., Hyun, J.M., 1996. Numerical simulations of a piston-type owmeter of high linearity. Flow Meas. Instrum. 7, 69–75. Wojtkowiak, J., Kim, W.N., Hyun, J.M., 1997. Computations of ow characteristics of a rotating-piston-type owmeter. Flow Meas. Instrum. 8, 27–37. Wu-Shung Fu, Ching-Chi Tseng, 1994. Enhancement of heat transfer for a tube with an inner tube insertion. Int. J. Heat Mass Transfer 37, 499–509.
Flow and heat transfer in a pipe containing a coaxially-rotating disk Janusz Wojtkowiak a , Jae Min Hyunb;∗ a
b
Institute of Environmental Engineering, PoznaÃn University of Technology, Piotrowo 3a, 60-965 PoznaÃn, Poland Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-ku, Taejon 305-701, South Korea Received 6 May 1999; received in revised form 2 July 1999; accepted 27 August 1999
Abstract A numerical investigation was made of steady laminar ow and convective heat transfer in a pipe constricted by a coaxially rotating disk. The analysis was executed by using the nite volume approach. Calculations were made for the through- ow Reynolds numbers (based on the pipe radius) of 10, 25, 50, 75, 100, 125, 150, the rotational Reynolds numbers of 0, 250, 500 and 1000, for the disk-to-pipe radius ratios of 0.9, 0.95 and 0.99, and for the Prandtl numbers of 0.7 and 7. The heat transfer rate, the pressure drop coecient, and the temperature distributions are determined. The results show that the temperature and ow characteristics are substantially a ected by the rotation of the disk. In the disk downstream- ow, the wall and disk recirculation zones are noticed. The swirl imparted to the ow measurably increases the heat transfer rate. The increase is especially noticeable when the wall recirculation region does not exist. Physical rationalizations are made of the computed ow features, and a brief description is given of the Nusselt number distribution. c 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
Keywords: High blockage ow; Heat transfer augmentation; Numerical simulation
1. Introduction Laminar ow of an incompressible viscous uid through a narrow annular passage in a pipe constitutes an essential building block of ow measurement technology and heat transfer enhancement techniques. The ow characteristics inside oating-element owmeters (rotameters), with a stationary or a rotating oat, were investigated numerically and experimentally by Buckle et al. (1992, 1995). ∗
Corresponding author. Tel.: +82-42-8693012; fax: +82-42-869-3210. E-mail addresses: [email protected] (J. Wojtkowiak), [email protected] (J.M. Hyun) c 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/00/$ 20.00 All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 9 9 ) 0 0 0 3 6 - 2
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Nomenclature A f l1 ; L1 l1 ; L2 ld ; Ld Min Nu Nuav p; P p; P Pr Q r; R rd ; Rd rp ; Rp Re Re! s t; T Tin t b ; Tb Tw u; U v; V w; W x y z; Z
uid thermal di usivity local pressure drop coecient distance between the inlet of the pipe and the disk upstream surface, ll = Ll =Rp distance between the downstream surface of the disk and the pipe outlet, l2 = L2 =Rp disk thickness, ld = Ld =Rp mean axial velocity of the uid, =Q=(R2p ) local Nusselt number average Nusselt number pressure, p = P=(Min2 ) pressure di erence between the inlet and outlet of the pipe, p = P=(Min2 ) Prandtl number, =vf =A volume ow rate radial coordinates, r = R=Rp disk radius, rd = Rd =Rp pipe radius, rp = Rp =Rp = 1 through- ow Reynolds number, =Min Rp =vf rotational Reynolds number, =!R2p =vf axial location of the wall separation point temperature, t = T=Tin temperature of the uid at the pipe inlet
uid bulk temperature, t b = Tb =Tin pipe wall temperature velocity components in the axial direction, u = U=Min velocity components in the radial direction, v = V=Min velocity components in the azimuthal direction, w = W=Min length of the disk recirculation region, x = X=Rp length of the wall recirculation regions, y = Y=Rp axial coordinates, z = Z=Rp
Greek letters
uid density vf
uid kinematic viscosity ! angular velocity of the disk A new type of piston- owmeter was proposed by Peters and Kuralt (1995), in which the uid moves through the narrow gap between the lateral surface of the piston and the inner surface of the pipe wall. Numerical evaluations of the system geometries, ow patterns and pressure distributions inside these owmeters with stationary and rotating pistons were conducted by Wojtkowiak et al. (1996, 1997). Several methods of heat transfer augmentation in laminar pipe ow have been summarized in the literature (Bergles, 1969, 1973, Kalinin et al., 1990). One popular method is to employ various types of concentric insertions (Rowley and Patankar, 1984; Vilemas et al., 1987; Budov and
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Dimitrev, 1989; Agrawal and Sengupta, 1989; Wu-Shung Fu and Ching-Chi Tseng, 1994), which generally demonstrates increased convective heat transports since swirl is imparted to the ow. The quantitative results are, however, not entirely consistent, and the detailed ow characteristics warrant an in-depth scrutiny. In the present paper, as a simpli ed paradigmatic model, the laminar ow passing through a circular pipe, which is constricted by a co-axially rotating disk, is considered. Comprehensive numerical computations are performed by solving the full Navier–Stokes equations. The ranges of parameters pertinent to practical applications are identi ed, and the local pressure elds, velocity pro les and Nusselt number distributions are exhibited. The global ow patterns are illustrated, and the behavior of secondary ow bubbles is depicted. The heat transfer rate, pressure drop coecient and temperature distributions are described. Emphasis is placed on delineating the changes in ow properties, which are brought forth by the rotation of the disk. The ranges of computational parameters are chosen as follows: the main- ow Reynolds number (based on the pipe radius) 106Re6150; the rotational Reynolds number of the disk 06Re! 61000; the Prandtl number Pr = 0:7 and 7; and the disk=pipe radius ratio rd = 0:9; 0:95 and 0.99. The thrust is given to the depiction and rationalization of the major ow features. A brief description will be made of the behavior of Nusselt numbers on the pipe wall. 2. Mathematical model and solution procedure Steady, axisymmetric, incompressible, laminar ow of a Newtonian uid is considered. The thermophysical properties of the uid are taken to be constant. With these assumptions, using cylindrical coordinates (r; ; z), with corresponding velocity components (v; w; u), the governing equations, in non-dimensional form, are written as 1 @(rv) @u + = 0; r @r @z
(1)
@v w2 @p 1 @v =− + v +u − @r @z r @r Re
@2 v 1 @v @2 v v + 2− 2 + 2 @r r @r @z r
@2 u 1 @u @2 u + + 2 @r 2 r @r @z
@w @w vw 1 v +u + = @r @z r Re
@2 w 1 @w @2 w w + 2 − 2 + 2 @r r @r @z r
"
;
(2)
!
@p @u @u 1 v +u =− + @r @z @z Re
1 @ @t @t @t 1 v +u = r @r @z Re Pr r @r @r
!
;
(3) !
;
(4)
#
@2 t + 2 : @z
(5)
In the above, nondimensionalization has been implemented using the pipe radius Rp , the average inlet velocity Min , and the average inlet temperature di erence of the uid (Tin − Tw ) as the reference
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Fig. 1. Computational domain and boundary conditions.
length, velocity and temperature, respectively. Consequently, the dimensionless variables are de ned as R Z U V W P T − Tw r= ; z= ; u= ; v= ; w= ; p= ; t= ; Rp Rp Min Min Min Min2 Tin − Tw (6) R2p ! Rp Min vf Re = ; Re! = ; Pr = : vf vf A The actual ow con guration and the computational domain are outlined in Fig. 1. Considerations are given to the proper boundary conditions. At the pipe inlet, the ow is assumed to be hydraulically fully developed, with the average velocity equal to one, and with the radially uniform temperature (t = 1), i.e., u = 2(1 − r 2 );
v = w = 0;
t = 1;
at z = 0;
06r ¡ 1:
(7)
At the pipe outlet, the ow is fully developed, with zero gradients of ow variables, i.e., @u @v @w @t = = = = 0; at z = l1 + ld + l2 ; 06r ¡ 1: (8) @z @z @z @z On all the pipe wall, no-slip conditions for the velocities and a constant temperature are speci ed, i.e., u = v = w = t = 0;
at 06z6l1 + ld + l2 ; r = 1:
On the upstream and downstream surfaces of the disk, the conditions can be expressed as Re! u = v = t = 0; w = r at z = l1 ; 06r6rd and at z = l1 + ld ; 06r6rd : Re On the lateral surface of the disk, similar conditions are applied, Re! u = v = t = 0; w = rd ; at l1 6z6l1 + ld ; r = rd : Re Symmetry conditions are speci ed on the axis, i.e., @t @u @w = = v = 0; at r = 0: = @r @r @r
(9) (10)
(11)
(12)
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The discretized representations of the governing equations (1) – (5), together with the boundary conditions (7) – (12), were solved by utilizing the well documented nite-volume, iterative algorithm, SIMPLER, of Patankar (1980). Convective terms were discretized by using the reformulated QUICK scheme of Hayase et al. (1992). This provides improved physical consistency of the numerical solution than other methods (e.g., upwind or hybrid central=upwind). Moreover, the treatment of source terms by Hayase ensures the continuity of the ux through the boundary, which contributes to numerical stability. The central di erence scheme was used for the di usive terms. The upstream length l1 and downstream length l2 were selected to allow a sucient entry length near the inlet and to justify the zero-gradient condition at the outlet (see Fig. 1). A trial-and-error method was used to reach the values l1 and l2 . For most conditions, the upstream length l1 = 5, and the downstream length l2 = 55 were found to be adequate, except for the highest Reynolds numbers in the computations (Re = 150; Re! = 1000), for which a longer downstream length (l2 = 75) was adopted. This elongation was introduced in order to minimize the in uence of the outlet boundary condition and to assist in obtaining a converged solution. The thickness of the disk was taken to be ld = 0:005. Extremely large values (1030 ) of viscosity and thermal conductivity, along with zero source terms, were speci ed for all the points within the disk in order to simulate the no- ow and perfect conductor conditions there. A staggered grid was adopted and grid stretching was implemented. Extensive trial calculations were carried out to ascertain the grid-convergence characteristics of the results. These comprehensive sensitivity tests examined the impacts of the variations in the total number of cells, locations of concentrated cells and grid expansion ratios, among others. For the majority of the runs, the 40×160 grid in the r–z coordinates was found to be optimal in terms of overall computational eciency and accuracy. Only for the larger downstream length (l2 = 75), the 40 × 180 grid was deployed. The general grid arrangement was such that the grid points were concentrated close to the disk surface, the disk edge and the pipe wall. In the axial direction 20 cells were placed in the upstream region of the disk, and 3 cells for the disk and 137 (157 for l2 = 75) cells in the downstream region of the disk. In the radial direction 9,7 and 5 cells were placed in the gap for rd = 0:9; 0:95 and 0.99, respectively. In all the cases, the solution was declared to have converged when the global residuals of the dependent variables were less than 10−4 . In the course of analyzing the numerical results, it is advantageous to introduce the following physical quantities. The coecient of local pressure drop is f=
P − 32=Re(L1 + Ld + L2 )=2Rp 12 Min2 16 (l1 + ld + l2 ); = 2p − 1 2 Re Min 2
(13)
where, the overall pressure drop P = Pin − Pout , in which Pin and Pout denote the pressures at the pipe inlet and outlet, respectively. In Eq. (13), 32 L1 + Ld + L2 1 M 2 Re 2Rp 2 in indicates the pressure losses for Hagen–Poiseuille pipe ow. The local Nusselt number at the pipe wall is 2Rp @T 2 @t Nu = − =− ; Tb − Tw @R R=Rp t b @r r=1
(14)
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and the average Nusselt number is 1 Nuav = l1 + l d + l 2 is
Z
l1 +ld +l2
0
Nu d z:
(15)
In the above, the bulk temperature of the uid in the upstream and downstream sides of the disk 2 Tb = Min R2p
Z
Rp
0
UTR dR;
tb = 2
Z 0
1
utr dr;
(16)
and the bulk temperature in the gap region is 2 Tb; g = Min R2p
Z
Rp
Rd
UTR dR;
t b; g = 2
Z
1
Rd
utr dr:
(17)
3. Results and discussion Before the main computations were started, veri cation tests were executed. For the testing calculations, the 40 × 160 staggered and stretched grid was employed. First, a straight pipe of length ll + ld + l2 = 60, with no obstructions, was used. The results were highly consistent with the existing analytical and experimental data (e.g., Kays and Crawford 1993, White 1991). In terms of local and average Nusselt numbers, pressure drop coecient and entrance length, the discrepancies were smaller than 6%. Next, test runs were made to compute the local pressure drop coecient for a stationary disk in a pipe. For all the through- ow Reynolds numbers, the di erences between the computed results and the experimental data of Idelchik (1996) were less than 10% for rd = 0:9, 13% for rd = 0:95 and 20% for rd = 0:99. Finally, in order to assess the performance of the present calculations for ows with separation, the ow in rotameter was computed. The numerical results were in broad agreement with the published data of Buckle et al. (1992,1995). Plots of the numerically constructed meridional stream functions are exempli ed in Figs. 2 and 3. Fig. 2 shows the case of a stationary disk (Re! = 0, and rd = 0:95). As Re varies, the ow structure on the upstream-side of the disk is a ected in a minor way, whereas the ow on the downstream side undergoes conspicuous changes. For a small Reynolds number (see Fig. 2a for Re = 10), the ow patterns are qualitatively similar on both sides of the disk, i.e., no separation, and no recirculation zones are visible. Physically, for a very low Reynolds number, the inertial forces are insucient to induce separation, and a general pattern is akin to a creeping ow. As Re increases (see Fig. 2b for Re = 50), an annular wall-jet is formed, and one large recirculation region on the downstream face of the disk is visible. For larger Reynolds numbers (see Fig. 2c for Re = 100), a new recirculation region begins to form on the pipe wall due to the adverse pressure gradient caused by the sudden expansion. On the downstream side of the disk, the increasing pressure slows down the wall ow and pushes the uid to move backward. A qualitatively similar sequence of events is observed for rd = 0:9 and 0.99. As rd increases, the formation of the recirculation regions takes place earlier, i.e., the separation bubbles occur at lower Re values. Fig. 3 depicts a representative series of computational results for the case of a rotating disk (Re! =500, and rd =0:95). The rotating disk induces axial ows toward its upstream and downstream surfaces. The uid then moves radially outward on the surface of the disk, and it travels along the
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Fig. 2. Plots of the meridional stream function for the case rd = 0:95 and Re! = 0: (a) Re = 10; (b) Re = 50; (c) Re = 100.
Fig. 3. Plots of the meridional stream function for the case rd = 0:95 and Re! = 500: (a) Re = 10; (b) Re = 50; (c) Re = 100.
pipe wall, away from the disk. The interaction of this ow with the main through- ow produces a complex pattern, as exhibited in Fig. 3. At a small Reynolds number (see Fig. 3a for Re = 10), the existence of the recirculation regions is apparent on both the upstream and downstream sides of the disk. The upstream-side bubble is much smaller than the downstream side one. This is due to the fact that, on the upstream-side of the disk, the main through- ow near the pipe wall, and the pipe wall ow driven by the rotating disk are in the opposite direction. On the downstream side of the disk, the above two wall ows are in the same direction. At a slightly higher Re, the upstream-side recirculation bubble vanishes, and the global ow pattern is dominated by the main through- ow. As Re increases further (see Fig. 3b for Re = 50), a “two-celled” structure of the disk
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Fig. 4. Schematic of the principal ow elements.
recirculation region is visible in the meridional plane. A similar “two-celled” internal structure has been reported in swirling pipe ow undergoing vortex breakdown (see, e.g., Faler and Leibovitch, 1978; Escudier, 1988; Brown and Lopez, 1990, and more recently Keller, 1995). The present results display qualitatively similar patterns. However, no de nitive conclusions can be drawn at this point, since there are notable di erences in the ow geometry and boundary conditions. In the present problem, the main (outer) disk recirculation zone is due principally to the abrupt change in the
ow geometry. In the case of conventional vortex-breakdown bubble, the dynamic unbalance in the swirling uid motion is cited to be the dominant cause. As Re increases further (see Fig. 3c for Re=100), an elongated separation bubble is formed on the pipe wall, and a strong interaction between the disk and the wall recirculation regions is noticeable. The size of the wall bubble increases, and the length of the disk recirculation zone is reduced. Simultaneously, changes in the internal structure of the disk region are evident. The “two-celled” structure is preserved, but the internal bubble moves closer to the disk surface. The above-described sequence is qualitatively similar for other values of Re! and rd adopted in the study. The foregoing observations based on the numerically constructed visualizations are summarized to produce a quantitative portrayal of the two principal ow elements, i.e., the disk recirculation zone and the wall recirculation zone. Fig. 4 o ers a schema of the ow description; i.e., the length of the disk recirculation region (x), the length of the wall recirculation region (y), and the location of the wall separation point (s). For a stationary disk (Re! = 0), Figs. 5 and 6, respectively, show the x–Re and y–Re plots. When the gap clearance is extremely small, x decreases as Re increases. As stipulated earlier, when the wall recirculation region is formed, the disk recirculation zone shrinks in size. Also, the wall recirculation region grows relatively fast as Re or rd increases. However, in the parameter space in which no wall recirculation zone exists, x naturally increases with increasing Re, and the results in Fig. 5 re ect these qualitative features. Similarly, Fig. 6 illustrates that y increases as Re or rd increases. The mechanism of the formation of the wall recirculation zone is the presence of a positive (adverse) pressure gradient in the downstream side of the disk. As noted, the pressure gradient grows rapidly with Re and with rd , which is in line with the above physical explanation. Figs. 7 and 8 present the results for a rotating disk. As Re or rd increases, y increases, which is qualitatively akin to the case of a stationary disk (see Fig. 6). However, y decreases as Re! increases. A plausible physical argument is that, as the disk rotates, the azimuthal velocity component is developed. This causes a further reduction of pressure in the direction of main ow, which gives
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Fig. 5. Disk recirculation region length x vs. Reynolds number for a stationary disk (Re! = 0).
Fig. 6. Wall recirculation region length y vs. Reynolds number for a stationary disk (Re! = 0).
rise to a favorable (negative) pressure gradient. Also, it should be pointed out that the pressure drop in the gap contributed by the rotating disk is much smaller than the pressure drop caused by the through- ow in the narrow clearance. In accord with these physical considerations, y is reduced as Re! increases and as Re decreases. The qualitative trends detectable in Fig. 8 are compatible with the foregoing physical interpretations. The wall separation point moves closer to the disk as Re and rd increase and as Re! decreases. Physically, the adverse (positive) pressure gradient strengthens as Re and rd increase, which leads to a reduction in s. In contrast, as Re! increases, the favorable (negative) pressure gradient strengthens, which causes an increase in s. The plots in Figs. 5 and 8 indicate that s is generally smaller than x. This implies intense interactions between the separation process and the disk recirculation zone.
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Fig. 7. Wall recirculation region length y vs. Reynolds number for rd = 0:95.
Fig. 8. Wall separation point position s vs. Reynolds number.
In Fig. 9, the relationship between the local pressure drop coecient (f) (see Eq. (13)) and the Reynolds number (Re) is presented. The f–Re variation when Re is small and moderate is typical of laminar ow conditions, i.e., a steep decrease of f is noticeable as Re increases. The asymptotic value at large Re is characteristic of a turbulent ow, for which, as well known, the local pressure drop coecient is approximately independent of the Reynolds number. Fig. 9 also clearly demonstrates the obvious dependence of f upon rd and Re! . It is seen that f increases with rd and, to a considerably lower degree, with Re! . When Re and rd are large, the e ect of Re! on
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Fig. 9. Local pressure drop coecient f vs. Reynolds number: (a) rd = 0:9; (b) rd = 0:95; (c) rd = 0:99.
f is negligible. In such cases, the pressure drop in the main ow is much larger than the additional pressure decrease due to the disk rotation. Next, heat transport characteristics are mentioned. In Figs. 10 and 11, the axial pro les of local Nusselt number on the pipe wall as well as the variations of the uid bulk temperature are illustrated. The general shape of Nu-pro les remains largely unchanged. At the entrance (z = 0), a uniform temperature pro le is speci ed, which gives a large value of Nu. Afterwards the thermal boundary layer grows, and Nu decreases. As the uid reaches the disk region, the uid is accelerated. The
uid impinges on the cold pipe wall, which leads to a sudden and extremely localized increase in Nu. Downstream of this disk region, Nu generally drops o sharply to a very low value, which is caused by reduced uid velocities owing out to the pipe outlet. The distributions of the Nusselt numbers downstream of the disk are inspected. The distributions re ect the complex ow structure in the downstream region of the disk. In general, when swirl is imparted to the ow the Nusselt number increases (compare e.g. Figs. 10a with 10b for z ¿ 5). However, when the wall recirculation region begins to form, some decrease in Nu is visible (see Figs. 10b and 11b for Re = 100). This is due to the fact that the wall recirculation zone causes the main through- ow to move away from
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Fig. 10. Bulk temperature t and local Nusselt number Nu for rd = 0:95 and Pr = 7: (a) Re! = 0; (b) Re! = 500.
the pipe wall. It is worthwhile to note that the minimum value of the Nusselt number occurs in the vicinity of the separation point of the wall recirculation region. For downstream close to the exit, Nu tends to settle down to the value for a fully developed pipe ow. The above numerical data indicate that the e ects of the Re and especially of Re! on Nu, in the downstream region of the disk, are substantial. Obviously, as Re and Re! increase, convective activities become more vigorous, which increases Nu. The pro les of the bulk temperature t are also included in Figs. 10 and 11. A monotonic decrease in t is seen in the upstream region of the disk. In the disk gap region, a rapid decrease in t is visible. In the downstream region, a mild decrease in t is evident. In general, the rate of decrease in t becomes more pronounced as Re decreases, and Re! increases. Minor irregularities in the bulk temperature plots, which are noticeable on the disk upstream-side (e.g., Fig. 10b for Re = 10), and on the disk downstream-side (e.g., Fig. 10a for Re = 100), are attributable to the existence of the upstream side (Fig. 3a), and the downstream side (Fig. 3c) wall recirculation regions.
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Fig. 11. Same as in Fig. 10, except for Pr = 0:7.
A combination of the reverse ow in these regions with the main ow produces the bulk temperature peaks, as exhibited in Figs. 10 and 11. For instance, in the exemplary plots of Figs. 10 and 11, the t b -curves for Re = 50 very close to the disk are strongly a ected by the relative strengths of upstream- and downstream-side recirculation regions. As remarked previously, the existence and size of these recirculation regions are extremely sensitive to Re and other ow parameters. In view of the complexity of the problem formulation, parametric evaluations of t b -distributions with Re, Re! and rd are beyond the scope of the present endeavor. A precise examination of heat transfer characteristics can be made by deploying far more grid points, and this task will be tackled in the forthcoming papers. Finally, the average Nusselt number Nuav is plotted in Fig. 12. Obviously, Nuav increases with Re, Re! and rd . It is worthwhile to note that the increase in average Nusselt number due to the decrease of disk=pipe clearance (from rd = 0:9 to rd = 0:95) is relatively small, whereas the increase in Nuav , due to the disk rotation, is substantial.
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Fig. 12. Average Nusselt number Nu av vs. Reynolds number for Pr = 7: (a) rd = 0:9; (b) rd = 0:95.
4. Concluding remarks Numerical results have been obtained for steady laminar uid ow and heat transfer in a pipe with a coaxially rotating disk. It is shown that the rotation of the disk has profound impact on the
ow and heat transfer characteristics especially in the downstream side of the disk. A large value of disk=pipe radius ratio causes a high-pressure loss and leads to the formation of wall recirculation regions. These contribute to the lowering of Nu. The swirl imparted to the ow augments both heat transport and pressure loss, but the increase in pressure loss is limited.
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Acknowledgements Appreciation is extended to the referee whose constructive comments led to improvements in the revised manuscript. References Agrawal, A.K., Sengupta, S., 1989. Laminar ow and heat transfer in blocked annuli. Num. Heat Transfer (Part A) 15, 489–508. Bergles, A.E., 1969. Survey and evaluation of techniques to augment convective heat and mass transfer. In Progress in Heat and Mass Transfer. Pergamon Press, Oxford, Vol. 1, pp. 331– 424. Bergles, A.E., 1973. Recent developments in convective heat transfer augmentation. Appl. Mech. Rev. 26, 675– 682. Brown, G.L., Lopez, J.M., 1990. Axisymmetric vortex breakdown. Part 2. physical mechanisms. J. Fluid Mech. 221, 553–576. Budov, V.M., Dimitrev, S.M., 1989. Compact Heat Exchangers of Nuclear Power Stations. Energoatomizdat, Moskva (in Russian). Buckle, U., Durst, F., Howe, B., Melling, A., 1992. Investigation of a oating element owmeter. Flow Meas. Instrum. 3, 215–225. Buckle, U., Durst, F., Kochner, H., Melling, A., 1995. Further investigation of a oating element owmeter. Flow Meas. Instrum. 6, 75–78. Escudier, M., 1988. Vortex breakdown: observations and explanations. Prog. Aerospace Sci. 25, 189–229. Faler, J.H., Leibovitch, S., 1978. An experimental map of an internal structure of a vortex breakdown. J. Fluid Mech. 86, 313–335. Hayase, T., Humphrey, J.A.C., Greif, R.A., 1992. Consistently formulated QUICK scheme for fast and stable convergence using nite-volume iterative calculation procedures. J. Comput. Phys. 98, 108–118. Idelchik, A.V., 1996. Handbook of Hydraulic Resistance, 3rd Edition. Begell House, New York. Kalinin, E.K., Dreicer, G.A., Jarho, S.A., 1990. Heat Transfer Augmentation in Channels, 3rd Edition. Masinostroenie, Moskva (in Russian). Kays, W.M., Crawford, M.E., 1993. Convective Heat and Mass Transfer, 3rd Edition. McGraw-Hill, New York. Keller, J.J., 1995. On the interpretation of vortex breakdown. Phys. Fluids 7, 1695 –1702. Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York. Peters, F., Kuralt, T., 1995. A gas owmeter of high linearity. Flow Meas. Instrum. 6, 29–32. Rowley, G.J., Patankar, S.V., 1984. Analysis of laminar ow and heat transfer in tubs with internal circumferential ns. Int. J. Heat Mass Transfer 27, 553–560. Vilemas, J., Cesna, B., Survila, V., 1987. In: Zukauskas, A., Karni, J. (Eds.), Heat Transfer in Gas-Cooled Annular Channels. Hemisphere, Washington, DC. White, F.M., 1991. Viscous Fluid Flow, 2nd Edition. McGraw-Hill, New York. Wojtkowiak, J., Kim, W.N., Hyun, J.M., 1996. Numerical simulations of a piston-type owmeter of high linearity. Flow Meas. Instrum. 7, 69–75. Wojtkowiak, J., Kim, W.N., Hyun, J.M., 1997. Computations of ow characteristics of a rotating-piston-type owmeter. Flow Meas. Instrum. 8, 27–37. Wu-Shung Fu, Ching-Chi Tseng, 1994. Enhancement of heat transfer for a tube with an inner tube insertion. Int. J. Heat Mass Transfer 37, 499–509.